English
// libdivide.h
// Copyright 2010 - 2018 ridiculous_fish
//
// libdivide is dual-licensed under the Boost or zlib licenses.
// You may use libdivide under the terms of either of these.
// See LICENSE.txt for more details.
#ifndef LIBDIVIDE_H
#define LIBDIVIDE_H
#if defined(_MSC_VER)
// disable warning C4146: unary minus operator applied to
// unsigned type, result still unsigned
#pragma warning(disable: 4146)
#define LIBDIVIDE_VC
#endif
#ifdef __cplusplus
#include <cstdlib>
#include <cstdio>
#else
#include <stdlib.h>
#include <stdio.h>
#endif
#include <stdint.h>
#if defined(LIBDIVIDE_USE_SSE2)
#include <emmintrin.h>
#endif
#if defined(LIBDIVIDE_VC)
#include <intrin.h>
#endif
#ifndef __has_builtin
#define __has_builtin(x) 0 // Compatibility with non-clang compilers.
#endif
#if defined(__SIZEOF_INT128__)
#define HAS_INT128_T
#endif
#if defined(__x86_64__) || defined(_WIN64) || defined(_M_X64)
#define LIBDIVIDE_IS_X86_64
#endif
#if defined(__i386__)
#define LIBDIVIDE_IS_i386
#endif
#if defined(__GNUC__) || defined(__clang__)
#define LIBDIVIDE_GCC_STYLE_ASM
#endif
#if defined(__cplusplus) || defined(LIBDIVIDE_VC)
#define LIBDIVIDE_FUNCTION __FUNCTION__
#else
#define LIBDIVIDE_FUNCTION __func__
#endif
#define LIBDIVIDE_ERROR(msg) \
do { \
fprintf(stderr, "libdivide.h:%d: %s(): Error: %s\n", \
__LINE__, LIBDIVIDE_FUNCTION, msg); \
exit(-1); \
} while (0)
#if defined(LIBDIVIDE_ASSERTIONS_ON)
#define LIBDIVIDE_ASSERT(x) \
do { \
if (!(x)) { \
fprintf(stderr, "libdivide.h:%d: %s(): Assertion failed: %s\n", \
__LINE__, LIBDIVIDE_FUNCTION, #x); \
exit(-1); \
} \
} while (0)
#else
#define LIBDIVIDE_ASSERT(x)
#endif
// libdivide may use the pmuldq (vector signed 32x32->64 mult instruction)
// which is in SSE 4.1. However, signed multiplication can be emulated
// efficiently with unsigned multiplication, and SSE 4.1 is currently rare, so
// it is OK to not turn this on.
#ifdef LIBDIVIDE_USE_SSE4_1
#include <smmintrin.h>
#endif
#ifdef __cplusplus
// We place libdivide within the libdivide namespace, and that goes in an
// anonymous namespace so that the functions are only visible to files that
// #include this header and don't get external linkage. At least that's the
// theory.
namespace {
namespace libdivide {
#endif
// Explanation of "more" field: bit 6 is whether to use shift path. If we are
// using the shift path, bit 7 is whether the divisor is negative in the signed
// case; in the unsigned case it is 0. Bits 0-4 is shift value (for shift
// path or mult path). In 32 bit case, bit 5 is always 0. We use bit 7 as the
// "negative divisor indicator" so that we can use sign extension to
// efficiently go to a full-width -1.
//
// u32: [0-4] shift value
// [5] ignored
// [6] add indicator
// [7] shift path
//
// s32: [0-4] shift value
// [5] shift path
// [6] add indicator
// [7] indicates negative divisor
//
// u64: [0-5] shift value
// [6] add indicator
// [7] shift path
//
// s64: [0-5] shift value
// [6] add indicator
// [7] indicates negative divisor
// magic number of 0 indicates shift path (we ran out of bits!)
//
// In s32 and s64 branchfree modes, the magic number is negated according to
// whether the divisor is negated. In branchfree strategy, it is not negated.
enum {
LIBDIVIDE_32_SHIFT_MASK = 0x1F,
LIBDIVIDE_64_SHIFT_MASK = 0x3F,
LIBDIVIDE_ADD_MARKER = 0x40,
LIBDIVIDE_U32_SHIFT_PATH = 0x80,
LIBDIVIDE_U64_SHIFT_PATH = 0x80,
LIBDIVIDE_S32_SHIFT_PATH = 0x20,
LIBDIVIDE_NEGATIVE_DIVISOR = 0x80
};
// pack divider structs to prevent compilers from padding.
// This reduces memory usage by up to 43% when using a large
// array of libdivide dividers and improves performance
// by up to 10% because of reduced memory bandwidth.
#pragma pack(push, 1)
struct libdivide_u32_t {
uint32_t magic;
uint8_t more;
};
struct libdivide_s32_t {
int32_t magic;
uint8_t more;
};
struct libdivide_u64_t {
uint64_t magic;
uint8_t more;
};
struct libdivide_s64_t {
int64_t magic;
uint8_t more;
};
struct libdivide_u32_branchfree_t {
uint32_t magic;
uint8_t more;
};
struct libdivide_s32_branchfree_t {
int32_t magic;
uint8_t more;
};
struct libdivide_u64_branchfree_t {
uint64_t magic;
uint8_t more;
};
struct libdivide_s64_branchfree_t {
int64_t magic;
uint8_t more;
};
#pragma pack(pop)
#ifndef LIBDIVIDE_API
#ifdef __cplusplus
// In C++, we don't want our public functions to be static, because
// they are arguments to templates and static functions can't do that.
// They get internal linkage through virtue of the anonymous namespace.
// In C, they should be static.
#define LIBDIVIDE_API
#else
#define LIBDIVIDE_API static inline
#endif
#endif
LIBDIVIDE_API struct libdivide_s32_t libdivide_s32_gen(int32_t y);
LIBDIVIDE_API struct libdivide_u32_t libdivide_u32_gen(uint32_t y);
LIBDIVIDE_API struct libdivide_s64_t libdivide_s64_gen(int64_t y);
LIBDIVIDE_API struct libdivide_u64_t libdivide_u64_gen(uint64_t y);
LIBDIVIDE_API struct libdivide_s32_branchfree_t libdivide_s32_branchfree_gen(int32_t y);
LIBDIVIDE_API struct libdivide_u32_branchfree_t libdivide_u32_branchfree_gen(uint32_t y);
LIBDIVIDE_API struct libdivide_s64_branchfree_t libdivide_s64_branchfree_gen(int64_t y);
LIBDIVIDE_API struct libdivide_u64_branchfree_t libdivide_u64_branchfree_gen(uint64_t y);
LIBDIVIDE_API int32_t libdivide_s32_do(int32_t numer, const struct libdivide_s32_t *denom);
LIBDIVIDE_API uint32_t libdivide_u32_do(uint32_t numer, const struct libdivide_u32_t *denom);
LIBDIVIDE_API int64_t libdivide_s64_do(int64_t numer, const struct libdivide_s64_t *denom);
LIBDIVIDE_API uint64_t libdivide_u64_do(uint64_t y, const struct libdivide_u64_t *denom);
LIBDIVIDE_API int32_t libdivide_s32_branchfree_do(int32_t numer, const struct libdivide_s32_branchfree_t *denom);
LIBDIVIDE_API uint32_t libdivide_u32_branchfree_do(uint32_t numer, const struct libdivide_u32_branchfree_t *denom);
LIBDIVIDE_API int64_t libdivide_s64_branchfree_do(int64_t numer, const struct libdivide_s64_branchfree_t *denom);
LIBDIVIDE_API uint64_t libdivide_u64_branchfree_do(uint64_t y, const struct libdivide_u64_branchfree_t *denom);
LIBDIVIDE_API int32_t libdivide_s32_recover(const struct libdivide_s32_t *denom);
LIBDIVIDE_API uint32_t libdivide_u32_recover(const struct libdivide_u32_t *denom);
LIBDIVIDE_API int64_t libdivide_s64_recover(const struct libdivide_s64_t *denom);
LIBDIVIDE_API uint64_t libdivide_u64_recover(const struct libdivide_u64_t *denom);
LIBDIVIDE_API int32_t libdivide_s32_branchfree_recover(const struct libdivide_s32_branchfree_t *denom);
LIBDIVIDE_API uint32_t libdivide_u32_branchfree_recover(const struct libdivide_u32_branchfree_t *denom);
LIBDIVIDE_API int64_t libdivide_s64_branchfree_recover(const struct libdivide_s64_branchfree_t *denom);
LIBDIVIDE_API uint64_t libdivide_u64_branchfree_recover(const struct libdivide_u64_branchfree_t *denom);
LIBDIVIDE_API int libdivide_u32_get_algorithm(const struct libdivide_u32_t *denom);
LIBDIVIDE_API uint32_t libdivide_u32_do_alg0(uint32_t numer, const struct libdivide_u32_t *denom);
LIBDIVIDE_API uint32_t libdivide_u32_do_alg1(uint32_t numer, const struct libdivide_u32_t *denom);
LIBDIVIDE_API uint32_t libdivide_u32_do_alg2(uint32_t numer, const struct libdivide_u32_t *denom);
LIBDIVIDE_API int libdivide_u64_get_algorithm(const struct libdivide_u64_t *denom);
LIBDIVIDE_API uint64_t libdivide_u64_do_alg0(uint64_t numer, const struct libdivide_u64_t *denom);
LIBDIVIDE_API uint64_t libdivide_u64_do_alg1(uint64_t numer, const struct libdivide_u64_t *denom);
LIBDIVIDE_API uint64_t libdivide_u64_do_alg2(uint64_t numer, const struct libdivide_u64_t *denom);
LIBDIVIDE_API int libdivide_s32_get_algorithm(const struct libdivide_s32_t *denom);
LIBDIVIDE_API int32_t libdivide_s32_do_alg0(int32_t numer, const struct libdivide_s32_t *denom);
LIBDIVIDE_API int32_t libdivide_s32_do_alg1(int32_t numer, const struct libdivide_s32_t *denom);
LIBDIVIDE_API int32_t libdivide_s32_do_alg2(int32_t numer, const struct libdivide_s32_t *denom);
LIBDIVIDE_API int32_t libdivide_s32_do_alg3(int32_t numer, const struct libdivide_s32_t *denom);
LIBDIVIDE_API int32_t libdivide_s32_do_alg4(int32_t numer, const struct libdivide_s32_t *denom);
LIBDIVIDE_API int libdivide_s64_get_algorithm(const struct libdivide_s64_t *denom);
LIBDIVIDE_API int64_t libdivide_s64_do_alg0(int64_t numer, const struct libdivide_s64_t *denom);
LIBDIVIDE_API int64_t libdivide_s64_do_alg1(int64_t numer, const struct libdivide_s64_t *denom);
LIBDIVIDE_API int64_t libdivide_s64_do_alg2(int64_t numer, const struct libdivide_s64_t *denom);
LIBDIVIDE_API int64_t libdivide_s64_do_alg3(int64_t numer, const struct libdivide_s64_t *denom);
LIBDIVIDE_API int64_t libdivide_s64_do_alg4(int64_t numer, const struct libdivide_s64_t *denom);
#if defined(LIBDIVIDE_USE_SSE2)
LIBDIVIDE_API __m128i libdivide_u32_do_vector(__m128i numers, const struct libdivide_u32_t *denom);
LIBDIVIDE_API __m128i libdivide_s32_do_vector(__m128i numers, const struct libdivide_s32_t *denom);
LIBDIVIDE_API __m128i libdivide_u64_do_vector(__m128i numers, const struct libdivide_u64_t *denom);
LIBDIVIDE_API __m128i libdivide_s64_do_vector(__m128i numers, const struct libdivide_s64_t *denom);
LIBDIVIDE_API __m128i libdivide_u32_do_vector_alg0(__m128i numers, const struct libdivide_u32_t *denom);
LIBDIVIDE_API __m128i libdivide_u32_do_vector_alg1(__m128i numers, const struct libdivide_u32_t *denom);
LIBDIVIDE_API __m128i libdivide_u32_do_vector_alg2(__m128i numers, const struct libdivide_u32_t *denom);
LIBDIVIDE_API __m128i libdivide_s32_do_vector_alg0(__m128i numers, const struct libdivide_s32_t *denom);
LIBDIVIDE_API __m128i libdivide_s32_do_vector_alg1(__m128i numers, const struct libdivide_s32_t *denom);
LIBDIVIDE_API __m128i libdivide_s32_do_vector_alg2(__m128i numers, const struct libdivide_s32_t *denom);
LIBDIVIDE_API __m128i libdivide_s32_do_vector_alg3(__m128i numers, const struct libdivide_s32_t *denom);
LIBDIVIDE_API __m128i libdivide_s32_do_vector_alg4(__m128i numers, const struct libdivide_s32_t *denom);
LIBDIVIDE_API __m128i libdivide_u64_do_vector_alg0(__m128i numers, const struct libdivide_u64_t *denom);
LIBDIVIDE_API __m128i libdivide_u64_do_vector_alg1(__m128i numers, const struct libdivide_u64_t *denom);
LIBDIVIDE_API __m128i libdivide_u64_do_vector_alg2(__m128i numers, const struct libdivide_u64_t *denom);
LIBDIVIDE_API __m128i libdivide_s64_do_vector_alg0(__m128i numers, const struct libdivide_s64_t *denom);
LIBDIVIDE_API __m128i libdivide_s64_do_vector_alg1(__m128i numers, const struct libdivide_s64_t *denom);
LIBDIVIDE_API __m128i libdivide_s64_do_vector_alg2(__m128i numers, const struct libdivide_s64_t *denom);
LIBDIVIDE_API __m128i libdivide_s64_do_vector_alg3(__m128i numers, const struct libdivide_s64_t *denom);
LIBDIVIDE_API __m128i libdivide_s64_do_vector_alg4(__m128i numers, const struct libdivide_s64_t *denom);
LIBDIVIDE_API __m128i libdivide_u32_branchfree_do_vector(__m128i numers, const struct libdivide_u32_branchfree_t *denom);
LIBDIVIDE_API __m128i libdivide_s32_branchfree_do_vector(__m128i numers, const struct libdivide_s32_branchfree_t *denom);
LIBDIVIDE_API __m128i libdivide_u64_branchfree_do_vector(__m128i numers, const struct libdivide_u64_branchfree_t *denom);
LIBDIVIDE_API __m128i libdivide_s64_branchfree_do_vector(__m128i numers, const struct libdivide_s64_branchfree_t *denom);
#endif
//////// Internal Utility Functions
static inline uint32_t libdivide__mullhi_u32(uint32_t x, uint32_t y) {
uint64_t xl = x, yl = y;
uint64_t rl = xl * yl;
return (uint32_t)(rl >> 32);
}
static uint64_t libdivide__mullhi_u64(uint64_t x, uint64_t y) {
#if defined(LIBDIVIDE_VC) && defined(LIBDIVIDE_IS_X86_64)
return __umulh(x, y);
#elif defined(HAS_INT128_T)
__uint128_t xl = x, yl = y;
__uint128_t rl = xl * yl;
return (uint64_t)(rl >> 64);
#else
// full 128 bits are x0 * y0 + (x0 * y1 << 32) + (x1 * y0 << 32) + (x1 * y1 << 64)
uint32_t mask = 0xFFFFFFFF;
uint32_t x0 = (uint32_t)(x & mask);
uint32_t x1 = (uint32_t)(x >> 32);
uint32_t y0 = (uint32_t)(y & mask);
uint32_t y1 = (uint32_t)(y >> 32);
uint32_t x0y0_hi = libdivide__mullhi_u32(x0, y0);
uint64_t x0y1 = x0 * (uint64_t)y1;
uint64_t x1y0 = x1 * (uint64_t)y0;
uint64_t x1y1 = x1 * (uint64_t)y1;
uint64_t temp = x1y0 + x0y0_hi;
uint64_t temp_lo = temp & mask;
uint64_t temp_hi = temp >> 32;
return x1y1 + temp_hi + ((temp_lo + x0y1) >> 32);
#endif
}
static inline int64_t libdivide__mullhi_s64(int64_t x, int64_t y) {
#if defined(LIBDIVIDE_VC) && defined(LIBDIVIDE_IS_X86_64)
return __mulh(x, y);
#elif defined(HAS_INT128_T)
__int128_t xl = x, yl = y;
__int128_t rl = xl * yl;
return (int64_t)(rl >> 64);
#else
// full 128 bits are x0 * y0 + (x0 * y1 << 32) + (x1 * y0 << 32) + (x1 * y1 << 64)
uint32_t mask = 0xFFFFFFFF;
uint32_t x0 = (uint32_t)(x & mask);
uint32_t y0 = (uint32_t)(y & mask);
int32_t x1 = (int32_t)(x >> 32);
int32_t y1 = (int32_t)(y >> 32);
uint32_t x0y0_hi = libdivide__mullhi_u32(x0, y0);
int64_t t = x1 * (int64_t)y0 + x0y0_hi;
int64_t w1 = x0 * (int64_t)y1 + (t & mask);
return x1 * (int64_t)y1 + (t >> 32) + (w1 >> 32);
#endif
}
#if defined(LIBDIVIDE_USE_SSE2)
static inline __m128i libdivide__u64_to_m128(uint64_t x) {
#if defined(LIBDIVIDE_VC) && !defined(_WIN64)
// 64 bit windows doesn't seem to have an implementation of any of these
// load intrinsics, and 32 bit Visual C++ crashes
_declspec(align(16)) uint64_t temp[2] = {x, x};
return _mm_load_si128((const __m128i*)temp);
#else
// everyone else gets it right
return _mm_set1_epi64x(x);
#endif
}
static inline __m128i libdivide_get_FFFFFFFF00000000(void) {
// returns the same as _mm_set1_epi64(0xFFFFFFFF00000000ULL)
// without touching memory.
// optimizes to pcmpeqd on OS X
__m128i result = _mm_set1_epi8(-1);
return _mm_slli_epi64(result, 32);
}
static inline __m128i libdivide_get_00000000FFFFFFFF(void) {
// returns the same as _mm_set1_epi64(0x00000000FFFFFFFFULL)
// without touching memory.
// optimizes to pcmpeqd on OS X
__m128i result = _mm_set1_epi8(-1);
result = _mm_srli_epi64(result, 32);
return result;
}
static inline __m128i libdivide_s64_signbits(__m128i v) {
// we want to compute v >> 63, that is, _mm_srai_epi64(v, 63). But there
// is no 64 bit shift right arithmetic instruction in SSE2. So we have to
// fake it by first duplicating the high 32 bit values, and then using a 32
// bit shift. Another option would be to use _mm_srli_epi64(v, 63) and
// then subtract that from 0, but that approach appears to be substantially
// slower for unknown reasons
__m128i hiBitsDuped = _mm_shuffle_epi32(v, _MM_SHUFFLE(3, 3, 1, 1));
__m128i signBits = _mm_srai_epi32(hiBitsDuped, 31);
return signBits;
}
// Returns an __m128i whose low 32 bits are equal to amt and has zero elsewhere.
static inline __m128i libdivide_u32_to_m128i(uint32_t amt) {
return _mm_set_epi32(0, 0, 0, amt);
}
static inline __m128i libdivide_s64_shift_right_vector(__m128i v, int amt) {
// implementation of _mm_sra_epi64. Here we have two 64 bit values which
// are shifted right to logically become (64 - amt) values, and are then
// sign extended from a (64 - amt) bit number.
const int b = 64 - amt;
__m128i m = libdivide__u64_to_m128(1ULL << (b - 1));
__m128i x = _mm_srl_epi64(v, libdivide_u32_to_m128i(amt));
__m128i result = _mm_sub_epi64(_mm_xor_si128(x, m), m); // result = x^m - m
return result;
}
// Here, b is assumed to contain one 32 bit value repeated four times.
// If it did not, the function would not work.
static inline __m128i libdivide__mullhi_u32_flat_vector(__m128i a, __m128i b) {
__m128i hi_product_0Z2Z = _mm_srli_epi64(_mm_mul_epu32(a, b), 32);
__m128i a1X3X = _mm_srli_epi64(a, 32);
__m128i mask = libdivide_get_FFFFFFFF00000000();
__m128i hi_product_Z1Z3 = _mm_and_si128(_mm_mul_epu32(a1X3X, b), mask);
return _mm_or_si128(hi_product_0Z2Z, hi_product_Z1Z3); // = hi_product_0123
}
// Here, y is assumed to contain one 64 bit value repeated twice.
static inline __m128i libdivide_mullhi_u64_flat_vector(__m128i x, __m128i y) {
// full 128 bits are x0 * y0 + (x0 * y1 << 32) + (x1 * y0 << 32) + (x1 * y1 << 64)
__m128i mask = libdivide_get_00000000FFFFFFFF();
// x0 is low half of 2 64 bit values, x1 is high half in low slots
__m128i x0 = _mm_and_si128(x, mask);
__m128i x1 = _mm_srli_epi64(x, 32);
__m128i y0 = _mm_and_si128(y, mask);
__m128i y1 = _mm_srli_epi64(y, 32);
// x0 happens to have the low half of the two 64 bit values in 32 bit slots
// 0 and 2, so _mm_mul_epu32 computes their full product, and then we shift
// right by 32 to get just the high values
__m128i x0y0_hi = _mm_srli_epi64(_mm_mul_epu32(x0, y0), 32);
__m128i x0y1 = _mm_mul_epu32(x0, y1);
__m128i x1y0 = _mm_mul_epu32(x1, y0);
__m128i x1y1 = _mm_mul_epu32(x1, y1);
__m128i temp = _mm_add_epi64(x1y0, x0y0_hi);
__m128i temp_lo = _mm_and_si128(temp, mask);
__m128i temp_hi = _mm_srli_epi64(temp, 32);
temp_lo = _mm_srli_epi64(_mm_add_epi64(temp_lo, x0y1), 32);
temp_hi = _mm_add_epi64(x1y1, temp_hi);
return _mm_add_epi64(temp_lo, temp_hi);
}
// y is one 64 bit value repeated twice
static inline __m128i libdivide_mullhi_s64_flat_vector(__m128i x, __m128i y) {
__m128i p = libdivide_mullhi_u64_flat_vector(x, y);
__m128i t1 = _mm_and_si128(libdivide_s64_signbits(x), y);
p = _mm_sub_epi64(p, t1);
__m128i t2 = _mm_and_si128(libdivide_s64_signbits(y), x);
p = _mm_sub_epi64(p, t2);
return p;
}
#ifdef LIBDIVIDE_USE_SSE4_1
// b is one 32 bit value repeated four times.
static inline __m128i libdivide_mullhi_s32_flat_vector(__m128i a, __m128i b) {
__m128i hi_product_0Z2Z = _mm_srli_epi64(_mm_mul_epi32(a, b), 32);
__m128i a1X3X = _mm_srli_epi64(a, 32);
__m128i mask = libdivide_get_FFFFFFFF00000000();
__m128i hi_product_Z1Z3 = _mm_and_si128(_mm_mul_epi32(a1X3X, b), mask);
return _mm_or_si128(hi_product_0Z2Z, hi_product_Z1Z3); // = hi_product_0123
}
#else
// SSE2 does not have a signed multiplication instruction, but we can convert
// unsigned to signed pretty efficiently. Again, b is just a 32 bit value
// repeated four times.
static inline __m128i libdivide_mullhi_s32_flat_vector(__m128i a, __m128i b) {
__m128i p = libdivide__mullhi_u32_flat_vector(a, b);
__m128i t1 = _mm_and_si128(_mm_srai_epi32(a, 31), b); // t1 = (a >> 31) & y, arithmetic shift
__m128i t2 = _mm_and_si128(_mm_srai_epi32(b, 31), a);
p = _mm_sub_epi32(p, t1);
p = _mm_sub_epi32(p, t2);
return p;
}
#endif // LIBDIVIDE_USE_SSE4_1
#endif // LIBDIVIDE_USE_SSE2
static inline int32_t libdivide__count_leading_zeros32(uint32_t val) {
#if defined(__GNUC__) || __has_builtin(__builtin_clz)
// Fast way to count leading zeros
return __builtin_clz(val);
#elif defined(LIBDIVIDE_VC)
unsigned long result;
if (_BitScanReverse(&result, val)) {
return 31 - result;
}
return 0;
#else
int32_t result = 0;
uint32_t hi = 1U << 31;
while (~val & hi) {
hi >>= 1;
result++;
}
return result;
#endif
}
static inline int32_t libdivide__count_leading_zeros64(uint64_t val) {
#if defined(__GNUC__) || __has_builtin(__builtin_clzll)
// Fast way to count leading zeros
return __builtin_clzll(val);
#elif defined(LIBDIVIDE_VC) && defined(_WIN64)
unsigned long result;
if (_BitScanReverse64(&result, val)) {
return 63 - result;
}
return 0;
#else
uint32_t hi = val >> 32;
uint32_t lo = val & 0xFFFFFFFF;
if (hi != 0) return libdivide__count_leading_zeros32(hi);
return 32 + libdivide__count_leading_zeros32(lo);
#endif
}
#if (defined(LIBDIVIDE_IS_i386) || defined(LIBDIVIDE_IS_X86_64)) && \
defined(LIBDIVIDE_GCC_STYLE_ASM)
// libdivide_64_div_32_to_32: divides a 64 bit uint {u1, u0} by a 32 bit
// uint {v}. The result must fit in 32 bits.
// Returns the quotient directly and the remainder in *r
static uint32_t libdivide_64_div_32_to_32(uint32_t u1, uint32_t u0, uint32_t v, uint32_t *r) {
uint32_t result;
__asm__("divl %[v]"
: "=a"(result), "=d"(*r)
: [v] "r"(v), "a"(u0), "d"(u1)
);
return result;
}
#else
static uint32_t libdivide_64_div_32_to_32(uint32_t u1, uint32_t u0, uint32_t v, uint32_t *r) {
uint64_t n = (((uint64_t)u1) << 32) | u0;
uint32_t result = (uint32_t)(n / v);
*r = (uint32_t)(n - result * (uint64_t)v);
return result;
}
#endif
#if defined(LIBDIVIDE_IS_X86_64) && \
defined(LIBDIVIDE_GCC_STYLE_ASM)
static uint64_t libdivide_128_div_64_to_64(uint64_t u1, uint64_t u0, uint64_t v, uint64_t *r) {
// u0 -> rax
// u1 -> rdx
// divq
uint64_t result;
__asm__("divq %[v]"
: "=a"(result), "=d"(*r)
: [v] "r"(v), "a"(u0), "d"(u1)
);
return result;
}
#else
// Code taken from Hacker's Delight:
// http://www.hackersdelight.org/HDcode/divlu.c.
// License permits inclusion here per:
// http://www.hackersdelight.org/permissions.htm
static uint64_t libdivide_128_div_64_to_64(uint64_t u1, uint64_t u0, uint64_t v, uint64_t *r) {
const uint64_t b = (1ULL << 32); // Number base (16 bits)
uint64_t un1, un0; // Norm. dividend LSD's
uint64_t vn1, vn0; // Norm. divisor digits
uint64_t q1, q0; // Quotient digits
uint64_t un64, un21, un10; // Dividend digit pairs
uint64_t rhat; // A remainder
int32_t s; // Shift amount for norm
// If overflow, set rem. to an impossible value,
// and return the largest possible quotient
if (u1 >= v) {
if (r != NULL)
*r = (uint64_t) -1;
return (uint64_t) -1;
}
// count leading zeros
s = libdivide__count_leading_zeros64(v);
if (s > 0) {
// Normalize divisor
v = v << s;
un64 = (u1 << s) | ((u0 >> (64 - s)) & (-s >> 31));
un10 = u0 << s; // Shift dividend left
} else {
// Avoid undefined behavior
un64 = u1 | u0;
un10 = u0;
}
// Break divisor up into two 32-bit digits
vn1 = v >> 32;
vn0 = v & 0xFFFFFFFF;
// Break right half of dividend into two digits
un1 = un10 >> 32;
un0 = un10 & 0xFFFFFFFF;
// Compute the first quotient digit, q1
q1 = un64 / vn1;
rhat = un64 - q1 * vn1;
while (q1 >= b || q1 * vn0 > b * rhat + un1) {
q1 = q1 - 1;
rhat = rhat + vn1;
if (rhat >= b)
break;
}
// Multiply and subtract
un21 = un64 * b + un1 - q1 * v;
// Compute the second quotient digit
q0 = un21 / vn1;
rhat = un21 - q0 * vn1;
while (q0 >= b || q0 * vn0 > b * rhat + un0) {
q0 = q0 - 1;
rhat = rhat + vn1;
if (rhat >= b)
break;
}
// If remainder is wanted, return it
if (r != NULL)
*r = (un21 * b + un0 - q0 * v) >> s;
return q1 * b + q0;
}
#endif
// Bitshift a u128 in place, left (signed_shift > 0) or right (signed_shift < 0)
static inline void libdivide_u128_shift(uint64_t *u1, uint64_t *u0, int32_t signed_shift)
{
if (signed_shift > 0) {
uint32_t shift = signed_shift;
*u1 <<= shift;
*u1 |= *u0 >> (64 - shift);
*u0 <<= shift;
} else {
uint32_t shift = -signed_shift;
*u0 >>= shift;
*u0 |= *u1 << (64 - shift);
*u1 >>= shift;
}
}
// Computes a 128 / 128 -> 64 bit division, with a 128 bit remainder.
static uint64_t libdivide_128_div_128_to_64(uint64_t u_hi, uint64_t u_lo, uint64_t v_hi, uint64_t v_lo, uint64_t *r_hi, uint64_t *r_lo) {
#if defined(HAS_INT128_T)
__uint128_t ufull = u_hi;
__uint128_t vfull = v_hi;
ufull = (ufull << 64) | u_lo;
vfull = (vfull << 64) | v_lo;
uint64_t res = (uint64_t)(ufull / vfull);
__uint128_t remainder = ufull - (vfull * res);
*r_lo = (uint64_t)remainder;
*r_hi = (uint64_t)(remainder >> 64);
return res;
#else
// Adapted from "Unsigned Doubleword Division" in Hacker's Delight
// We want to compute u / v
typedef struct { uint64_t hi; uint64_t lo; } u128_t;
u128_t u = {u_hi, u_lo};
u128_t v = {v_hi, v_lo};
if (v.hi == 0) {
// divisor v is a 64 bit value, so we just need one 128/64 division
// Note that we are simpler than Hacker's Delight here, because we know
// the quotient fits in 64 bits whereas Hacker's Delight demands a full
// 128 bit quotient
*r_hi = 0;
return libdivide_128_div_64_to_64(u.hi, u.lo, v.lo, r_lo);
}
// Here v >= 2**64
// We know that v.hi != 0, so count leading zeros is OK
// We have 0 <= n <= 63
uint32_t n = libdivide__count_leading_zeros64(v.hi);
// Normalize the divisor so its MSB is 1
u128_t v1t = v;
libdivide_u128_shift(&v1t.hi, &v1t.lo, n);
uint64_t v1 = v1t.hi; // i.e. v1 = v1t >> 64
// To ensure no overflow
u128_t u1 = u;
libdivide_u128_shift(&u1.hi, &u1.lo, -1);
// Get quotient from divide unsigned insn.
uint64_t rem_ignored;
uint64_t q1 = libdivide_128_div_64_to_64(u1.hi, u1.lo, v1, &rem_ignored);
// Undo normalization and division of u by 2.
u128_t q0 = {0, q1};
libdivide_u128_shift(&q0.hi, &q0.lo, n);
libdivide_u128_shift(&q0.hi, &q0.lo, -63);
// Make q0 correct or too small by 1
// Equivalent to `if (q0 != 0) q0 = q0 - 1;`
if (q0.hi != 0 || q0.lo != 0) {
q0.hi -= (q0.lo == 0); // borrow
q0.lo -= 1;
}
// Now q0 is correct.
// Compute q0 * v as q0v
// = (q0.hi << 64 + q0.lo) * (v.hi << 64 + v.lo)
// = (q0.hi * v.hi << 128) + (q0.hi * v.lo << 64) +
// (q0.lo * v.hi << 64) + q0.lo * v.lo)
// Each term is 128 bit
// High half of full product (upper 128 bits!) are dropped
u128_t q0v = {0, 0};
q0v.hi = q0.hi*v.lo + q0.lo*v.hi + libdivide__mullhi_u64(q0.lo, v.lo);
q0v.lo = q0.lo*v.lo;
// Compute u - q0v as u_q0v
// This is the remainder
u128_t u_q0v = u;
u_q0v.hi -= q0v.hi + (u.lo < q0v.lo); // second term is borrow
u_q0v.lo -= q0v.lo;
// Check if u_q0v >= v
// This checks if our remainder is larger than the divisor
if ((u_q0v.hi > v.hi) ||
(u_q0v.hi == v.hi && u_q0v.lo >= v.lo)) {
// Increment q0
q0.lo += 1;
q0.hi += (q0.lo == 0); // carry
// Subtract v from remainder
u_q0v.hi -= v.hi + (u_q0v.lo < v.lo);
u_q0v.lo -= v.lo;
}
*r_hi = u_q0v.hi;
*r_lo = u_q0v.lo;
LIBDIVIDE_ASSERT(q0.hi == 0);
return q0.lo;
#endif
}
////////// UINT32
static inline struct libdivide_u32_t libdivide_internal_u32_gen(uint32_t d, int branchfree) {
if (d == 0) {
LIBDIVIDE_ERROR("divider must be != 0");
}
struct libdivide_u32_t result;
uint32_t floor_log_2_d = 31 - libdivide__count_leading_zeros32(d);
if ((d & (d - 1)) == 0) {
// Power of 2
if (! branchfree) {
result.magic = 0;
result.more = floor_log_2_d | LIBDIVIDE_U32_SHIFT_PATH;
} else {
// We want a magic number of 2**32 and a shift of floor_log_2_d
// but one of the shifts is taken up by LIBDIVIDE_ADD_MARKER,
// so we subtract 1 from the shift
result.magic = 0;
result.more = (floor_log_2_d-1) | LIBDIVIDE_ADD_MARKER;
}
} else {
uint8_t more;
uint32_t rem, proposed_m;
proposed_m = libdivide_64_div_32_to_32(1U << floor_log_2_d, 0, d, &rem);
LIBDIVIDE_ASSERT(rem > 0 && rem < d);
const uint32_t e = d - rem;
// This power works if e < 2**floor_log_2_d.
if (!branchfree && (e < (1U << floor_log_2_d))) {
// This power works
more = floor_log_2_d;
} else {
// We have to use the general 33-bit algorithm. We need to compute
// (2**power) / d. However, we already have (2**(power-1))/d and
// its remainder. By doubling both, and then correcting the
// remainder, we can compute the larger division.
// don't care about overflow here - in fact, we expect it
proposed_m += proposed_m;
const uint32_t twice_rem = rem + rem;
if (twice_rem >= d || twice_rem < rem) proposed_m += 1;
more = floor_log_2_d | LIBDIVIDE_ADD_MARKER;
}
result.magic = 1 + proposed_m;
result.more = more;
// result.more's shift should in general be ceil_log_2_d. But if we
// used the smaller power, we subtract one from the shift because we're
// using the smaller power. If we're using the larger power, we
// subtract one from the shift because it's taken care of by the add
// indicator. So floor_log_2_d happens to be correct in both cases.
}
return result;
}
struct libdivide_u32_t libdivide_u32_gen(uint32_t d) {
return libdivide_internal_u32_gen(d, 0);
}
struct libdivide_u32_branchfree_t libdivide_u32_branchfree_gen(uint32_t d) {
if (d == 1) {
LIBDIVIDE_ERROR("branchfree divider must be != 1");
}
struct libdivide_u32_t tmp = libdivide_internal_u32_gen(d, 1);
struct libdivide_u32_branchfree_t ret = {tmp.magic, (uint8_t)(tmp.more & LIBDIVIDE_32_SHIFT_MASK)};
return ret;
}
uint32_t libdivide_u32_do(uint32_t numer, const struct libdivide_u32_t *denom) {
uint8_t more = denom->more;
if (more & LIBDIVIDE_U32_SHIFT_PATH) {
return numer >> (more & LIBDIVIDE_32_SHIFT_MASK);
}
else {
uint32_t q = libdivide__mullhi_u32(denom->magic, numer);
if (more & LIBDIVIDE_ADD_MARKER) {
uint32_t t = ((numer - q) >> 1) + q;
return t >> (more & LIBDIVIDE_32_SHIFT_MASK);
}
else {
// all upper bits are 0 - don't need to mask them off
return q >> more;
}
}
}
uint32_t libdivide_u32_recover(const struct libdivide_u32_t *denom) {
uint8_t more = denom->more;
uint8_t shift = more & LIBDIVIDE_32_SHIFT_MASK;
if (more & LIBDIVIDE_U32_SHIFT_PATH) {
return 1U << shift;
} else if (!(more & LIBDIVIDE_ADD_MARKER)) {
// We compute q = n/d = n*m / 2^(32 + shift)
// Therefore we have d = 2^(32 + shift) / m
// We need to ceil it.
// We know d is not a power of 2, so m is not a power of 2,
// so we can just add 1 to the floor
uint32_t hi_dividend = 1U << shift;
uint32_t rem_ignored;
return 1 + libdivide_64_div_32_to_32(hi_dividend, 0, denom->magic, &rem_ignored);
} else {
// Here we wish to compute d = 2^(32+shift+1)/(m+2^32).
// Notice (m + 2^32) is a 33 bit number. Use 64 bit division for now
// Also note that shift may be as high as 31, so shift + 1 will
// overflow. So we have to compute it as 2^(32+shift)/(m+2^32), and
// then double the quotient and remainder.
uint64_t half_n = 1ULL << (32 + shift);
uint64_t d = (1ULL << 32) | denom->magic;
// Note that the quotient is guaranteed <= 32 bits, but the remainder
// may need 33!
uint32_t half_q = (uint32_t)(half_n / d);
uint64_t rem = half_n % d;
// We computed 2^(32+shift)/(m+2^32)
// Need to double it, and then add 1 to the quotient if doubling th
// remainder would increase the quotient.
// Note that rem<<1 cannot overflow, since rem < d and d is 33 bits
uint32_t full_q = half_q + half_q + ((rem<<1) >= d);
// We rounded down in gen unless we're a power of 2 (i.e. in branchfree case)
// We can detect that by looking at m. If m zero, we're a power of 2
return full_q + (denom->magic != 0);
}
}
uint32_t libdivide_u32_branchfree_recover(const struct libdivide_u32_branchfree_t *denom) {
struct libdivide_u32_t denom_u32 = {denom->magic, (uint8_t)(denom->more | LIBDIVIDE_ADD_MARKER)};
return libdivide_u32_recover(&denom_u32);
}
int libdivide_u32_get_algorithm(const struct libdivide_u32_t *denom) {
uint8_t more = denom->more;
if (more & LIBDIVIDE_U32_SHIFT_PATH) return 0;
else if (!(more & LIBDIVIDE_ADD_MARKER)) return 1;
else return 2;
}
uint32_t libdivide_u32_do_alg0(uint32_t numer, const struct libdivide_u32_t *denom) {
return numer >> (denom->more & LIBDIVIDE_32_SHIFT_MASK);
}
uint32_t libdivide_u32_do_alg1(uint32_t numer, const struct libdivide_u32_t *denom) {
uint32_t q = libdivide__mullhi_u32(denom->magic, numer);
return q >> denom->more;
}
uint32_t libdivide_u32_do_alg2(uint32_t numer, const struct libdivide_u32_t *denom) {
// denom->add != 0
uint32_t q = libdivide__mullhi_u32(denom->magic, numer);
uint32_t t = ((numer - q) >> 1) + q;
// Note that this mask is typically free. Only the low bits are meaningful
// to a shift, so compilers can optimize out this AND.
return t >> (denom->more & LIBDIVIDE_32_SHIFT_MASK);
}
// same as algo 2
uint32_t libdivide_u32_branchfree_do(uint32_t numer, const struct libdivide_u32_branchfree_t *denom) {
uint32_t q = libdivide__mullhi_u32(denom->magic, numer);
uint32_t t = ((numer - q) >> 1) + q;
return t >> denom->more;
}
#if defined(LIBDIVIDE_USE_SSE2)
__m128i libdivide_u32_do_vector(__m128i numers, const struct libdivide_u32_t *denom) {
uint8_t more = denom->more;
if (more & LIBDIVIDE_U32_SHIFT_PATH) {
return _mm_srl_epi32(numers, libdivide_u32_to_m128i(more & LIBDIVIDE_32_SHIFT_MASK));
}
else {
__m128i q = libdivide__mullhi_u32_flat_vector(numers, _mm_set1_epi32(denom->magic));
if (more & LIBDIVIDE_ADD_MARKER) {
// uint32_t t = ((numer - q) >> 1) + q;
// return t >> denom->shift;
__m128i t = _mm_add_epi32(_mm_srli_epi32(_mm_sub_epi32(numers, q), 1), q);
return _mm_srl_epi32(t, libdivide_u32_to_m128i(more & LIBDIVIDE_32_SHIFT_MASK));
}
else {
// q >> denom->shift
return _mm_srl_epi32(q, libdivide_u32_to_m128i(more));
}
}
}
__m128i libdivide_u32_do_vector_alg0(__m128i numers, const struct libdivide_u32_t *denom) {
return _mm_srl_epi32(numers, libdivide_u32_to_m128i(denom->more & LIBDIVIDE_32_SHIFT_MASK));
}
__m128i libdivide_u32_do_vector_alg1(__m128i numers, const struct libdivide_u32_t *denom) {
__m128i q = libdivide__mullhi_u32_flat_vector(numers, _mm_set1_epi32(denom->magic));
return _mm_srl_epi32(q, libdivide_u32_to_m128i(denom->more));
}
__m128i libdivide_u32_do_vector_alg2(__m128i numers, const struct libdivide_u32_t *denom) {
__m128i q = libdivide__mullhi_u32_flat_vector(numers, _mm_set1_epi32(denom->magic));
__m128i t = _mm_add_epi32(_mm_srli_epi32(_mm_sub_epi32(numers, q), 1), q);
return _mm_srl_epi32(t, libdivide_u32_to_m128i(denom->more & LIBDIVIDE_32_SHIFT_MASK));
}
// same as algo 2
LIBDIVIDE_API __m128i libdivide_u32_branchfree_do_vector(__m128i numers, const struct libdivide_u32_branchfree_t *denom) {
__m128i q = libdivide__mullhi_u32_flat_vector(numers, _mm_set1_epi32(denom->magic));
__m128i t = _mm_add_epi32(_mm_srli_epi32(_mm_sub_epi32(numers, q), 1), q);
return _mm_srl_epi32(t, libdivide_u32_to_m128i(denom->more));
}
#endif
/////////// UINT64
static inline struct libdivide_u64_t libdivide_internal_u64_gen(uint64_t d, int branchfree) {
if (d == 0) {
LIBDIVIDE_ERROR("divider must be != 0");
}
struct libdivide_u64_t result;
uint32_t floor_log_2_d = 63 - libdivide__count_leading_zeros64(d);
if ((d & (d - 1)) == 0) {
// Power of 2
if (! branchfree) {
result.magic = 0;
result.more = floor_log_2_d | LIBDIVIDE_U64_SHIFT_PATH;
} else {
// We want a magic number of 2**64 and a shift of floor_log_2_d
// but one of the shifts is taken up by LIBDIVIDE_ADD_MARKER,
// so we subtract 1 from the shift
result.magic = 0;
result.more = (floor_log_2_d-1) | LIBDIVIDE_ADD_MARKER;
}
} else {
uint64_t proposed_m, rem;
uint8_t more;
// (1 << (64 + floor_log_2_d)) / d
proposed_m = libdivide_128_div_64_to_64(1ULL << floor_log_2_d, 0, d, &rem);
LIBDIVIDE_ASSERT(rem > 0 && rem < d);
const uint64_t e = d - rem;
// This power works if e < 2**floor_log_2_d.
if (!branchfree && e < (1ULL << floor_log_2_d)) {
// This power works
more = floor_log_2_d;
} else {
// We have to use the general 65-bit algorithm. We need to compute
// (2**power) / d. However, we already have (2**(power-1))/d and
// its remainder. By doubling both, and then correcting the
// remainder, we can compute the larger division.
// don't care about overflow here - in fact, we expect it
proposed_m += proposed_m;
const uint64_t twice_rem = rem + rem;
if (twice_rem >= d || twice_rem < rem) proposed_m += 1;
more = floor_log_2_d | LIBDIVIDE_ADD_MARKER;
}
result.magic = 1 + proposed_m;
result.more = more;
// result.more's shift should in general be ceil_log_2_d. But if we
// used the smaller power, we subtract one from the shift because we're
// using the smaller power. If we're using the larger power, we
// subtract one from the shift because it's taken care of by the add
// indicator. So floor_log_2_d happens to be correct in both cases,
// which is why we do it outside of the if statement.
}
return result;
}
struct libdivide_u64_t libdivide_u64_gen(uint64_t d) {
return libdivide_internal_u64_gen(d, 0);
}
struct libdivide_u64_branchfree_t libdivide_u64_branchfree_gen(uint64_t d) {
if (d == 1) {
LIBDIVIDE_ERROR("branchfree divider must be != 1");
}
struct libdivide_u64_t tmp = libdivide_internal_u64_gen(d, 1);
struct libdivide_u64_branchfree_t ret = {tmp.magic, (uint8_t)(tmp.more & LIBDIVIDE_64_SHIFT_MASK)};
return ret;
}
uint64_t libdivide_u64_do(uint64_t numer, const struct libdivide_u64_t *denom) {
uint8_t more = denom->more;
if (more & LIBDIVIDE_U64_SHIFT_PATH) {
return numer >> (more & LIBDIVIDE_64_SHIFT_MASK);
}
else {
uint64_t q = libdivide__mullhi_u64(denom->magic, numer);
if (more & LIBDIVIDE_ADD_MARKER) {
uint64_t t = ((numer - q) >> 1) + q;
return t >> (more & LIBDIVIDE_64_SHIFT_MASK);
}
else {
// all upper bits are 0 - don't need to mask them off
return q >> more;
}
}
}
uint64_t libdivide_u64_recover(const struct libdivide_u64_t *denom) {
uint8_t more = denom->more;
uint8_t shift = more & LIBDIVIDE_64_SHIFT_MASK;
if (more & LIBDIVIDE_U64_SHIFT_PATH) {
return 1ULL << shift;
} else if (!(more & LIBDIVIDE_ADD_MARKER)) {
// We compute q = n/d = n*m / 2^(64 + shift)
// Therefore we have d = 2^(64 + shift) / m
// We need to ceil it.
// We know d is not a power of 2, so m is not a power of 2,
// so we can just add 1 to the floor
uint64_t hi_dividend = 1ULL << shift;
uint64_t rem_ignored;
return 1 + libdivide_128_div_64_to_64(hi_dividend, 0, denom->magic, &rem_ignored);
} else {
// Here we wish to compute d = 2^(64+shift+1)/(m+2^64).
// Notice (m + 2^64) is a 65 bit number. This gets hairy. See
// libdivide_u32_recover for more on what we do here.
// TODO: do something better than 128 bit math
// Hack: if d is not a power of 2, this is a 128/128->64 divide
// If d is a power of 2, this may be a bigger divide
// However we can optimize that easily
if (denom->magic == 0) {
// 2^(64 + shift + 1) / (2^64) == 2^(shift + 1)
return 1ULL << (shift + 1);
}
// Full n is a (potentially) 129 bit value
// half_n is a 128 bit value
// Compute the hi half of half_n. Low half is 0.
uint64_t half_n_hi = 1ULL << shift, half_n_lo = 0;
// d is a 65 bit value. The high bit is always set to 1.
const uint64_t d_hi = 1, d_lo = denom->magic;
// Note that the quotient is guaranteed <= 64 bits,
// but the remainder may need 65!
uint64_t r_hi, r_lo;
uint64_t half_q = libdivide_128_div_128_to_64(half_n_hi, half_n_lo, d_hi, d_lo, &r_hi, &r_lo);
// We computed 2^(64+shift)/(m+2^64)
// Double the remainder ('dr') and check if that is larger than d
// Note that d is a 65 bit value, so r1 is small and so r1 + r1 cannot
// overflow
uint64_t dr_lo = r_lo + r_lo;
uint64_t dr_hi = r_hi + r_hi + (dr_lo < r_lo); // last term is carry
int dr_exceeds_d = (dr_hi > d_hi) || (dr_hi == d_hi && dr_lo >= d_lo);
uint64_t full_q = half_q + half_q + (dr_exceeds_d ? 1 : 0);
return full_q + 1;
}
}
uint64_t libdivide_u64_branchfree_recover(const struct libdivide_u64_branchfree_t *denom) {
struct libdivide_u64_t denom_u64 = {denom->magic, (uint8_t)(denom->more | LIBDIVIDE_ADD_MARKER)};
return libdivide_u64_recover(&denom_u64);
}
int libdivide_u64_get_algorithm(const struct libdivide_u64_t *denom) {
uint8_t more = denom->more;
if (more & LIBDIVIDE_U64_SHIFT_PATH) return 0;
else if (!(more & LIBDIVIDE_ADD_MARKER)) return 1;
else return 2;
}
uint64_t libdivide_u64_do_alg0(uint64_t numer, const struct libdivide_u64_t *denom) {
return numer >> (denom->more & LIBDIVIDE_64_SHIFT_MASK);
}
uint64_t libdivide_u64_do_alg1(uint64_t numer, const struct libdivide_u64_t *denom) {
uint64_t q = libdivide__mullhi_u64(denom->magic, numer);
return q >> denom->more;
}
uint64_t libdivide_u64_do_alg2(uint64_t numer, const struct libdivide_u64_t *denom) {
uint64_t q = libdivide__mullhi_u64(denom->magic, numer);
uint64_t t = ((numer - q) >> 1) + q;
return t >> (denom->more & LIBDIVIDE_64_SHIFT_MASK);
}
// same as alg 2
uint64_t libdivide_u64_branchfree_do(uint64_t numer, const struct libdivide_u64_branchfree_t *denom) {
uint64_t q = libdivide__mullhi_u64(denom->magic, numer);
uint64_t t = ((numer - q) >> 1) + q;
return t >> denom->more;
}
#if defined(LIBDIVIDE_USE_SSE2)
__m128i libdivide_u64_do_vector(__m128i numers, const struct libdivide_u64_t *denom) {
uint8_t more = denom->more;
if (more & LIBDIVIDE_U64_SHIFT_PATH) {
return _mm_srl_epi64(numers, libdivide_u32_to_m128i(more & LIBDIVIDE_64_SHIFT_MASK));
}
else {
__m128i q = libdivide_mullhi_u64_flat_vector(numers, libdivide__u64_to_m128(denom->magic));
if (more & LIBDIVIDE_ADD_MARKER) {
// uint32_t t = ((numer - q) >> 1) + q;
// return t >> denom->shift;
__m128i t = _mm_add_epi64(_mm_srli_epi64(_mm_sub_epi64(numers, q), 1), q);
return _mm_srl_epi64(t, libdivide_u32_to_m128i(more & LIBDIVIDE_64_SHIFT_MASK));
}
else {
// q >> denom->shift
return _mm_srl_epi64(q, libdivide_u32_to_m128i(more));
}
}
}
__m128i libdivide_u64_do_vector_alg0(__m128i numers, const struct libdivide_u64_t *denom) {
return _mm_srl_epi64(numers, libdivide_u32_to_m128i(denom->more & LIBDIVIDE_64_SHIFT_MASK));
}
__m128i libdivide_u64_do_vector_alg1(__m128i numers, const struct libdivide_u64_t *denom) {
__m128i q = libdivide_mullhi_u64_flat_vector(numers, libdivide__u64_to_m128(denom->magic));
return _mm_srl_epi64(q, libdivide_u32_to_m128i(denom->more));
}
__m128i libdivide_u64_do_vector_alg2(__m128i numers, const struct libdivide_u64_t *denom) {
__m128i q = libdivide_mullhi_u64_flat_vector(numers, libdivide__u64_to_m128(denom->magic));
__m128i t = _mm_add_epi64(_mm_srli_epi64(_mm_sub_epi64(numers, q), 1), q);
return _mm_srl_epi64(t, libdivide_u32_to_m128i(denom->more & LIBDIVIDE_64_SHIFT_MASK));
}
__m128i libdivide_u64_branchfree_do_vector(__m128i numers, const struct libdivide_u64_branchfree_t *denom) {
__m128i q = libdivide_mullhi_u64_flat_vector(numers, libdivide__u64_to_m128(denom->magic));
__m128i t = _mm_add_epi64(_mm_srli_epi64(_mm_sub_epi64(numers, q), 1), q);
return _mm_srl_epi64(t, libdivide_u32_to_m128i(denom->more));
}
#endif
/////////// SINT32
static inline int32_t libdivide__mullhi_s32(int32_t x, int32_t y) {
int64_t xl = x, yl = y;
int64_t rl = xl * yl;
// needs to be arithmetic shift
return (int32_t)(rl >> 32);
}
static inline struct libdivide_s32_t libdivide_internal_s32_gen(int32_t d, int branchfree) {
if (d == 0) {
LIBDIVIDE_ERROR("divider must be != 0");
}
struct libdivide_s32_t result;
// If d is a power of 2, or negative a power of 2, we have to use a shift.
// This is especially important because the magic algorithm fails for -1.
// To check if d is a power of 2 or its inverse, it suffices to check
// whether its absolute value has exactly one bit set. This works even for
// INT_MIN, because abs(INT_MIN) == INT_MIN, and INT_MIN has one bit set
// and is a power of 2.
uint32_t ud = (uint32_t)d;
uint32_t absD = (d < 0) ? -ud : ud;
uint32_t floor_log_2_d = 31 - libdivide__count_leading_zeros32(absD);
// check if exactly one bit is set,
// don't care if absD is 0 since that's divide by zero
if ((absD & (absD - 1)) == 0) {
// Branchfree and normal paths are exactly the same
result.magic = 0;
result.more = floor_log_2_d | (d < 0 ? LIBDIVIDE_NEGATIVE_DIVISOR : 0) | LIBDIVIDE_S32_SHIFT_PATH;
} else {
LIBDIVIDE_ASSERT(floor_log_2_d >= 1);
uint8_t more;
// the dividend here is 2**(floor_log_2_d + 31), so the low 32 bit word
// is 0 and the high word is floor_log_2_d - 1
uint32_t rem, proposed_m;
proposed_m = libdivide_64_div_32_to_32(1U << (floor_log_2_d - 1), 0, absD, &rem);
const uint32_t e = absD - rem;
// We are going to start with a power of floor_log_2_d - 1.
// This works if works if e < 2**floor_log_2_d.
if (!branchfree && e < (1U << floor_log_2_d)) {
// This power works
more = floor_log_2_d - 1;
} else {
// We need to go one higher. This should not make proposed_m
// overflow, but it will make it negative when interpreted as an
// int32_t.
proposed_m += proposed_m;
const uint32_t twice_rem = rem + rem;
if (twice_rem >= absD || twice_rem < rem) proposed_m += 1;
more = floor_log_2_d | LIBDIVIDE_ADD_MARKER;
}
proposed_m += 1;
int32_t magic = (int32_t)proposed_m;
// Mark if we are negative. Note we only negate the magic number in the
// branchfull case.
if (d < 0) {
more |= LIBDIVIDE_NEGATIVE_DIVISOR;
if (!branchfree) {
magic = -magic;
}
}
result.more = more;
result.magic = magic;
}
return result;
}
LIBDIVIDE_API struct libdivide_s32_t libdivide_s32_gen(int32_t d) {
return libdivide_internal_s32_gen(d, 0);
}
LIBDIVIDE_API struct libdivide_s32_branchfree_t libdivide_s32_branchfree_gen(int32_t d) {
if (d == 1) {
LIBDIVIDE_ERROR("branchfree divider must be != 1");
}
if (d == -1) {
LIBDIVIDE_ERROR("branchfree divider must be != -1");
}
struct libdivide_s32_t tmp = libdivide_internal_s32_gen(d, 1);
struct libdivide_s32_branchfree_t result = {tmp.magic, tmp.more};
return result;
}
int32_t libdivide_s32_do(int32_t numer, const struct libdivide_s32_t *denom) {
uint8_t more = denom->more;
if (more & LIBDIVIDE_S32_SHIFT_PATH) {
uint32_t sign = (int8_t)more >> 7;
uint8_t shifter = more & LIBDIVIDE_32_SHIFT_MASK;
uint32_t uq = (uint32_t)(numer + ((numer >> 31) & ((1U << shifter) - 1)));
int32_t q = (int32_t)uq;
q = q >> shifter;
q = (q ^ sign) - sign;
return q;
} else {
uint32_t uq = (uint32_t)libdivide__mullhi_s32(denom->magic, numer);
if (more & LIBDIVIDE_ADD_MARKER) {
// must be arithmetic shift and then sign extend
int32_t sign = (int8_t)more >> 7;
// q += (more < 0 ? -numer : numer), casts to avoid UB
uq += ((uint32_t)numer ^ sign) - sign;
}
int32_t q = (int32_t)uq;
q >>= more & LIBDIVIDE_32_SHIFT_MASK;
q += (q < 0);
return q;
}
}
int32_t libdivide_s32_branchfree_do(int32_t numer, const struct libdivide_s32_branchfree_t *denom) {
uint8_t more = denom->more;
uint8_t shift = more & LIBDIVIDE_32_SHIFT_MASK;
// must be arithmetic shift and then sign extend
int32_t sign = (int8_t)more >> 7;
int32_t magic = denom->magic;
int32_t q = libdivide__mullhi_s32(magic, numer);
q += numer;
// If q is non-negative, we have nothing to do
// If q is negative, we want to add either (2**shift)-1 if d is a power of
// 2, or (2**shift) if it is not a power of 2
uint32_t is_power_of_2 = !!(more & LIBDIVIDE_S32_SHIFT_PATH);
uint32_t q_sign = (uint32_t)(q >> 31);
q += q_sign & ((1 << shift) - is_power_of_2);
// Now arithmetic right shift
q >>= shift;
// Negate if needed
q = (q ^ sign) - sign;
return q;
}
int32_t libdivide_s32_recover(const struct libdivide_s32_t *denom) {
uint8_t more = denom->more;
uint8_t shift = more & LIBDIVIDE_32_SHIFT_MASK;
if (more & LIBDIVIDE_S32_SHIFT_PATH) {
uint32_t absD = 1U << shift;
if (more & LIBDIVIDE_NEGATIVE_DIVISOR) {
absD = -absD;
}
return (int32_t)absD;
} else {
// Unsigned math is much easier
// We negate the magic number only in the branchfull case, and we don't
// know which case we're in. However we have enough information to
// determine the correct sign of the magic number. The divisor was
// negative if LIBDIVIDE_NEGATIVE_DIVISOR is set. If ADD_MARKER is set,
// the magic number's sign is opposite that of the divisor.
// We want to compute the positive magic number.
int negative_divisor = (more & LIBDIVIDE_NEGATIVE_DIVISOR);
int magic_was_negated = (more & LIBDIVIDE_ADD_MARKER)
? denom->magic > 0 : denom->magic < 0;
// Handle the power of 2 case (including branchfree)
if (denom->magic == 0) {
int32_t result = 1 << shift;
return negative_divisor ? -result : result;
}
uint32_t d = (uint32_t)(magic_was_negated ? -denom->magic : denom->magic);
uint64_t n = 1ULL << (32 + shift); // this shift cannot exceed 30
uint32_t q = (uint32_t)(n / d);
int32_t result = (int32_t)q;
result += 1;
return negative_divisor ? -result : result;
}
}
int32_t libdivide_s32_branchfree_recover(const struct libdivide_s32_branchfree_t *denom) {
return libdivide_s32_recover((const struct libdivide_s32_t *)denom);
}
int libdivide_s32_get_algorithm(const struct libdivide_s32_t *denom) {
uint8_t more = denom->more;
int positiveDivisor = !(more & LIBDIVIDE_NEGATIVE_DIVISOR);
if (more & LIBDIVIDE_S32_SHIFT_PATH) return (positiveDivisor ? 0 : 1);
else if (more & LIBDIVIDE_ADD_MARKER) return (positiveDivisor ? 2 : 3);
else return 4;
}
int32_t libdivide_s32_do_alg0(int32_t numer, const struct libdivide_s32_t *denom) {
uint8_t shifter = denom->more & LIBDIVIDE_32_SHIFT_MASK;
int32_t q = numer + ((numer >> 31) & ((1U << shifter) - 1));
return q >> shifter;
}
int32_t libdivide_s32_do_alg1(int32_t numer, const struct libdivide_s32_t *denom) {
uint8_t shifter = denom->more & LIBDIVIDE_32_SHIFT_MASK;
int32_t q = numer + ((numer >> 31) & ((1U << shifter) - 1));
return - (q >> shifter);
}
int32_t libdivide_s32_do_alg2(int32_t numer, const struct libdivide_s32_t *denom) {
int32_t q = libdivide__mullhi_s32(denom->magic, numer);
q += numer;
q >>= denom->more & LIBDIVIDE_32_SHIFT_MASK;
q += (q < 0);
return q;
}
int32_t libdivide_s32_do_alg3(int32_t numer, const struct libdivide_s32_t *denom) {
int32_t q = libdivide__mullhi_s32(denom->magic, numer);
q -= numer;
q >>= denom->more & LIBDIVIDE_32_SHIFT_MASK;
q += (q < 0);
return q;
}
int32_t libdivide_s32_do_alg4(int32_t numer, const struct libdivide_s32_t *denom) {
int32_t q = libdivide__mullhi_s32(denom->magic, numer);
q >>= denom->more & LIBDIVIDE_32_SHIFT_MASK;
q += (q < 0);
return q;
}
#if defined(LIBDIVIDE_USE_SSE2)
__m128i libdivide_s32_do_vector(__m128i numers, const struct libdivide_s32_t *denom) {
uint8_t more = denom->more;
if (more & LIBDIVIDE_S32_SHIFT_PATH) {
uint32_t shifter = more & LIBDIVIDE_32_SHIFT_MASK;
__m128i roundToZeroTweak = _mm_set1_epi32((1U << shifter) - 1); // could use _mm_srli_epi32 with an all -1 register
__m128i q = _mm_add_epi32(numers, _mm_and_si128(_mm_srai_epi32(numers, 31), roundToZeroTweak)); //q = numer + ((numer >> 31) & roundToZeroTweak);
q = _mm_sra_epi32(q, libdivide_u32_to_m128i(shifter)); // q = q >> shifter
__m128i shiftMask = _mm_set1_epi32((int32_t)((int8_t)more >> 7)); // set all bits of shift mask = to the sign bit of more
q = _mm_sub_epi32(_mm_xor_si128(q, shiftMask), shiftMask); // q = (q ^ shiftMask) - shiftMask;
return q;
}
else {
__m128i q = libdivide_mullhi_s32_flat_vector(numers, _mm_set1_epi32(denom->magic));
if (more & LIBDIVIDE_ADD_MARKER) {
__m128i sign = _mm_set1_epi32((int32_t)(int8_t)more >> 7); // must be arithmetic shift
q = _mm_add_epi32(q, _mm_sub_epi32(_mm_xor_si128(numers, sign), sign)); // q += ((numer ^ sign) - sign);
}
q = _mm_sra_epi32(q, libdivide_u32_to_m128i(more & LIBDIVIDE_32_SHIFT_MASK)); // q >>= shift
q = _mm_add_epi32(q, _mm_srli_epi32(q, 31)); // q += (q < 0)
return q;
}
}
__m128i libdivide_s32_do_vector_alg0(__m128i numers, const struct libdivide_s32_t *denom) {
uint8_t shifter = denom->more & LIBDIVIDE_32_SHIFT_MASK;
__m128i roundToZeroTweak = _mm_set1_epi32((1U << shifter) - 1);
__m128i q = _mm_add_epi32(numers, _mm_and_si128(_mm_srai_epi32(numers, 31), roundToZeroTweak));
return _mm_sra_epi32(q, libdivide_u32_to_m128i(shifter));
}
__m128i libdivide_s32_do_vector_alg1(__m128i numers, const struct libdivide_s32_t *denom) {
uint8_t shifter = denom->more & LIBDIVIDE_32_SHIFT_MASK;
__m128i roundToZeroTweak = _mm_set1_epi32((1U << shifter) - 1);
__m128i q = _mm_add_epi32(numers, _mm_and_si128(_mm_srai_epi32(numers, 31), roundToZeroTweak));
return _mm_sub_epi32(_mm_setzero_si128(), _mm_sra_epi32(q, libdivide_u32_to_m128i(shifter)));
}
__m128i libdivide_s32_do_vector_alg2(__m128i numers, const struct libdivide_s32_t *denom) {
__m128i q = libdivide_mullhi_s32_flat_vector(numers, _mm_set1_epi32(denom->magic));
q = _mm_add_epi32(q, numers);
q = _mm_sra_epi32(q, libdivide_u32_to_m128i(denom->more & LIBDIVIDE_32_SHIFT_MASK));
q = _mm_add_epi32(q, _mm_srli_epi32(q, 31));
return q;
}
__m128i libdivide_s32_do_vector_alg3(__m128i numers, const struct libdivide_s32_t *denom) {
__m128i q = libdivide_mullhi_s32_flat_vector(numers, _mm_set1_epi32(denom->magic));
q = _mm_sub_epi32(q, numers);
q = _mm_sra_epi32(q, libdivide_u32_to_m128i(denom->more & LIBDIVIDE_32_SHIFT_MASK));
q = _mm_add_epi32(q, _mm_srli_epi32(q, 31));
return q;
}
__m128i libdivide_s32_do_vector_alg4(__m128i numers, const struct libdivide_s32_t *denom) {
uint8_t more = denom->more;
__m128i q = libdivide_mullhi_s32_flat_vector(numers, _mm_set1_epi32(denom->magic));
q = _mm_sra_epi32(q, libdivide_u32_to_m128i(more & LIBDIVIDE_32_SHIFT_MASK)); //q >>= shift
q = _mm_add_epi32(q, _mm_srli_epi32(q, 31)); // q += (q < 0)
return q;
}
__m128i libdivide_s32_branchfree_do_vector(__m128i numers, const struct libdivide_s32_branchfree_t *denom) {
int32_t magic = denom->magic;
uint8_t more = denom->more;
uint8_t shift = more & LIBDIVIDE_32_SHIFT_MASK;
// must be arithmetic shift
__m128i sign = _mm_set1_epi32((int32_t)(int8_t)more >> 7);
// libdivide__mullhi_s32(numers, magic);
__m128i q = libdivide_mullhi_s32_flat_vector(numers, _mm_set1_epi32(magic));
q = _mm_add_epi32(q, numers); // q += numers
// If q is non-negative, we have nothing to do
// If q is negative, we want to add either (2**shift)-1 if d is a power of
// 2, or (2**shift) if it is not a power of 2
uint32_t is_power_of_2 = (magic == 0);
__m128i q_sign = _mm_srai_epi32(q, 31); // q_sign = q >> 31
__m128i mask = _mm_set1_epi32((1 << shift) - is_power_of_2);
q = _mm_add_epi32(q, _mm_and_si128(q_sign, mask)); // q = q + (q_sign & mask)
q = _mm_srai_epi32(q, shift); //q >>= shift
q = _mm_sub_epi32(_mm_xor_si128(q, sign), sign); // q = (q ^ sign) - sign
return q;
}
#endif
///////////// SINT64
static inline struct libdivide_s64_t libdivide_internal_s64_gen(int64_t d, int branchfree) {
if (d == 0) {
LIBDIVIDE_ERROR("divider must be != 0");
}
struct libdivide_s64_t result;
// If d is a power of 2, or negative a power of 2, we have to use a shift.
// This is especially important because the magic algorithm fails for -1.
// To check if d is a power of 2 or its inverse, it suffices to check
// whether its absolute value has exactly one bit set. This works even for
// INT_MIN, because abs(INT_MIN) == INT_MIN, and INT_MIN has one bit set
// and is a power of 2.
uint64_t ud = (uint64_t)d;
uint64_t absD = (d < 0) ? -ud : ud;
uint32_t floor_log_2_d = 63 - libdivide__count_leading_zeros64(absD);
// check if exactly one bit is set,
// don't care if absD is 0 since that's divide by zero
if ((absD & (absD - 1)) == 0) {
// Branchfree and non-branchfree cases are the same
result.magic = 0;
result.more = floor_log_2_d | (d < 0 ? LIBDIVIDE_NEGATIVE_DIVISOR : 0);
} else {
// the dividend here is 2**(floor_log_2_d + 63), so the low 64 bit word
// is 0 and the high word is floor_log_2_d - 1
uint8_t more;
uint64_t rem, proposed_m;
proposed_m = libdivide_128_div_64_to_64(1ULL << (floor_log_2_d - 1), 0, absD, &rem);
const uint64_t e = absD - rem;
// We are going to start with a power of floor_log_2_d - 1.
// This works if works if e < 2**floor_log_2_d.
if (!branchfree && e < (1ULL << floor_log_2_d)) {
// This power works
more = floor_log_2_d - 1;
} else {
// We need to go one higher. This should not make proposed_m
// overflow, but it will make it negative when interpreted as an
// int32_t.
proposed_m += proposed_m;
const uint64_t twice_rem = rem + rem;
if (twice_rem >= absD || twice_rem < rem) proposed_m += 1;
// note that we only set the LIBDIVIDE_NEGATIVE_DIVISOR bit if we
// also set ADD_MARKER this is an annoying optimization that
// enables algorithm #4 to avoid the mask. However we always set it
// in the branchfree case
more = floor_log_2_d | LIBDIVIDE_ADD_MARKER;
}
proposed_m += 1;
int64_t magic = (int64_t)proposed_m;
// Mark if we are negative
if (d < 0) {
more |= LIBDIVIDE_NEGATIVE_DIVISOR;
if (!branchfree) {
magic = -magic;
}
}
result.more = more;
result.magic = magic;
}
return result;
}
struct libdivide_s64_t libdivide_s64_gen(int64_t d) {
return libdivide_internal_s64_gen(d, 0);
}
struct libdivide_s64_branchfree_t libdivide_s64_branchfree_gen(int64_t d) {
if (d == 1) {
LIBDIVIDE_ERROR("branchfree divider must be != 1");
}
if (d == -1) {
LIBDIVIDE_ERROR("branchfree divider must be != -1");
}
struct libdivide_s64_t tmp = libdivide_internal_s64_gen(d, 1);
struct libdivide_s64_branchfree_t ret = {tmp.magic, tmp.more};
return ret;
}
int64_t libdivide_s64_do(int64_t numer, const struct libdivide_s64_t *denom) {
uint8_t more = denom->more;
int64_t magic = denom->magic;
if (magic == 0) { //shift path
uint32_t shifter = more & LIBDIVIDE_64_SHIFT_MASK;
uint64_t uq = (uint64_t)numer + ((numer >> 63) & ((1ULL << shifter) - 1));
int64_t q = (int64_t)uq;
q = q >> shifter;
// must be arithmetic shift and then sign-extend
int64_t shiftMask = (int8_t)more >> 7;
q = (q ^ shiftMask) - shiftMask;
return q;
} else {
uint64_t uq = (uint64_t)libdivide__mullhi_s64(magic, numer);
if (more & LIBDIVIDE_ADD_MARKER) {
// must be arithmetic shift and then sign extend
int64_t sign = (int8_t)more >> 7;
uq += ((uint64_t)numer ^ sign) - sign;
}
int64_t q = (int64_t)uq;
q >>= more & LIBDIVIDE_64_SHIFT_MASK;
q += (q < 0);
return q;
}
}
int64_t libdivide_s64_branchfree_do(int64_t numer, const struct libdivide_s64_branchfree_t *denom) {
uint8_t more = denom->more;
uint32_t shift = more & LIBDIVIDE_64_SHIFT_MASK;
// must be arithmetic shift and then sign extend
int64_t sign = (int8_t)more >> 7;
int64_t magic = denom->magic;
int64_t q = libdivide__mullhi_s64(magic, numer);
q += numer;
// If q is non-negative, we have nothing to do.
// If q is negative, we want to add either (2**shift)-1 if d is a power of
// 2, or (2**shift) if it is not a power of 2.
uint32_t is_power_of_2 = (magic == 0);
uint64_t q_sign = (uint64_t)(q >> 63);
q += q_sign & ((1ULL << shift) - is_power_of_2);
// Arithmetic right shift
q >>= shift;
// Negate if needed
q = (q ^ sign) - sign;
return q;
}
int64_t libdivide_s64_recover(const struct libdivide_s64_t *denom) {
uint8_t more = denom->more;
uint8_t shift = more & LIBDIVIDE_64_SHIFT_MASK;
if (denom->magic == 0) { // shift path
uint64_t absD = 1ULL << shift;
if (more & LIBDIVIDE_NEGATIVE_DIVISOR) {
absD = -absD;
}
return (int64_t)absD;
} else {
// Unsigned math is much easier
int negative_divisor = (more & LIBDIVIDE_NEGATIVE_DIVISOR);
int magic_was_negated = (more & LIBDIVIDE_ADD_MARKER)
? denom->magic > 0 : denom->magic < 0;
uint64_t d = (uint64_t)(magic_was_negated ? -denom->magic : denom->magic);
uint64_t n_hi = 1ULL << shift, n_lo = 0;
uint64_t rem_ignored;
uint64_t q = libdivide_128_div_64_to_64(n_hi, n_lo, d, &rem_ignored);
int64_t result = (int64_t)(q + 1);
if (negative_divisor) {
result = -result;
}
return result;
}
}
int64_t libdivide_s64_branchfree_recover(const struct libdivide_s64_branchfree_t *denom) {
return libdivide_s64_recover((const struct libdivide_s64_t *)denom);
}
int libdivide_s64_get_algorithm(const struct libdivide_s64_t *denom) {
uint8_t more = denom->more;
int positiveDivisor = !(more & LIBDIVIDE_NEGATIVE_DIVISOR);
if (denom->magic == 0) return (positiveDivisor ? 0 : 1); // shift path
else if (more & LIBDIVIDE_ADD_MARKER) return (positiveDivisor ? 2 : 3);
else return 4;
}
int64_t libdivide_s64_do_alg0(int64_t numer, const struct libdivide_s64_t *denom) {
uint32_t shifter = denom->more & LIBDIVIDE_64_SHIFT_MASK;
int64_t q = numer + ((numer >> 63) & ((1ULL << shifter) - 1));
return q >> shifter;
}
int64_t libdivide_s64_do_alg1(int64_t numer, const struct libdivide_s64_t *denom) {
// denom->shifter != -1 && demo->shiftMask != 0
uint32_t shifter = denom->more & LIBDIVIDE_64_SHIFT_MASK;
int64_t q = numer + ((numer >> 63) & ((1ULL << shifter) - 1));
return - (q >> shifter);
}
int64_t libdivide_s64_do_alg2(int64_t numer, const struct libdivide_s64_t *denom) {
int64_t q = libdivide__mullhi_s64(denom->magic, numer);
q += numer;
q >>= denom->more & LIBDIVIDE_64_SHIFT_MASK;
q += (q < 0);
return q;
}
int64_t libdivide_s64_do_alg3(int64_t numer, const struct libdivide_s64_t *denom) {
int64_t q = libdivide__mullhi_s64(denom->magic, numer);
q -= numer;
q >>= denom->more & LIBDIVIDE_64_SHIFT_MASK;
q += (q < 0);
return q;
}
int64_t libdivide_s64_do_alg4(int64_t numer, const struct libdivide_s64_t *denom) {
int64_t q = libdivide__mullhi_s64(denom->magic, numer);
q >>= denom->more & LIBDIVIDE_64_SHIFT_MASK;
q += (q < 0);
return q;
}
#if defined(LIBDIVIDE_USE_SSE2)
__m128i libdivide_s64_do_vector(__m128i numers, const struct libdivide_s64_t *denom) {
uint8_t more = denom->more;
int64_t magic = denom->magic;
if (magic == 0) { // shift path
uint32_t shifter = more & LIBDIVIDE_64_SHIFT_MASK;
__m128i roundToZeroTweak = libdivide__u64_to_m128((1ULL << shifter) - 1);
__m128i q = _mm_add_epi64(numers, _mm_and_si128(libdivide_s64_signbits(numers), roundToZeroTweak)); // q = numer + ((numer >> 63) & roundToZeroTweak);
q = libdivide_s64_shift_right_vector(q, shifter); // q = q >> shifter
__m128i shiftMask = _mm_set1_epi32((int32_t)((int8_t)more >> 7));
q = _mm_sub_epi64(_mm_xor_si128(q, shiftMask), shiftMask); // q = (q ^ shiftMask) - shiftMask;
return q;
}
else {
__m128i q = libdivide_mullhi_s64_flat_vector(numers, libdivide__u64_to_m128(magic));
if (more & LIBDIVIDE_ADD_MARKER) {
__m128i sign = _mm_set1_epi32((int32_t)((int8_t)more >> 7)); // must be arithmetic shift
q = _mm_add_epi64(q, _mm_sub_epi64(_mm_xor_si128(numers, sign), sign)); // q += ((numer ^ sign) - sign);
}
// q >>= denom->mult_path.shift
q = libdivide_s64_shift_right_vector(q, more & LIBDIVIDE_64_SHIFT_MASK);
q = _mm_add_epi64(q, _mm_srli_epi64(q, 63)); // q += (q < 0)
return q;
}
}
__m128i libdivide_s64_do_vector_alg0(__m128i numers, const struct libdivide_s64_t *denom) {
uint32_t shifter = denom->more & LIBDIVIDE_64_SHIFT_MASK;
__m128i roundToZeroTweak = libdivide__u64_to_m128((1ULL << shifter) - 1);
__m128i q = _mm_add_epi64(numers, _mm_and_si128(libdivide_s64_signbits(numers), roundToZeroTweak));
q = libdivide_s64_shift_right_vector(q, shifter);
return q;
}
__m128i libdivide_s64_do_vector_alg1(__m128i numers, const struct libdivide_s64_t *denom) {
uint32_t shifter = denom->more & LIBDIVIDE_64_SHIFT_MASK;
__m128i roundToZeroTweak = libdivide__u64_to_m128((1ULL << shifter) - 1);
__m128i q = _mm_add_epi64(numers, _mm_and_si128(libdivide_s64_signbits(numers), roundToZeroTweak));
q = libdivide_s64_shift_right_vector(q, shifter);
return _mm_sub_epi64(_mm_setzero_si128(), q);
}
__m128i libdivide_s64_do_vector_alg2(__m128i numers, const struct libdivide_s64_t *denom) {
__m128i q = libdivide_mullhi_s64_flat_vector(numers, libdivide__u64_to_m128(denom->magic));
q = _mm_add_epi64(q, numers);
q = libdivide_s64_shift_right_vector(q, denom->more & LIBDIVIDE_64_SHIFT_MASK);
q = _mm_add_epi64(q, _mm_srli_epi64(q, 63)); // q += (q < 0)
return q;
}
__m128i libdivide_s64_do_vector_alg3(__m128i numers, const struct libdivide_s64_t *denom) {
__m128i q = libdivide_mullhi_s64_flat_vector(numers, libdivide__u64_to_m128(denom->magic));
q = _mm_sub_epi64(q, numers);
q = libdivide_s64_shift_right_vector(q, denom->more & LIBDIVIDE_64_SHIFT_MASK);
q = _mm_add_epi64(q, _mm_srli_epi64(q, 63)); // q += (q < 0)
return q;
}
__m128i libdivide_s64_do_vector_alg4(__m128i numers, const struct libdivide_s64_t *denom) {
__m128i q = libdivide_mullhi_s64_flat_vector(numers, libdivide__u64_to_m128(denom->magic));
q = libdivide_s64_shift_right_vector(q, denom->more & LIBDIVIDE_64_SHIFT_MASK);
q = _mm_add_epi64(q, _mm_srli_epi64(q, 63));
return q;
}
__m128i libdivide_s64_branchfree_do_vector(__m128i numers, const struct libdivide_s64_branchfree_t *denom) {
int64_t magic = denom->magic;
uint8_t more = denom->more;
uint8_t shift = more & LIBDIVIDE_64_SHIFT_MASK;
// must be arithmetic shift
__m128i sign = _mm_set1_epi32((int32_t)(int8_t)more >> 7);
// libdivide__mullhi_s64(numers, magic);
__m128i q = libdivide_mullhi_s64_flat_vector(numers, libdivide__u64_to_m128(magic));
q = _mm_add_epi64(q, numers); // q += numers
// If q is non-negative, we have nothing to do.
// If q is negative, we want to add either (2**shift)-1 if d is a power of
// 2, or (2**shift) if it is not a power of 2.
uint32_t is_power_of_2 = (magic == 0);
__m128i q_sign = libdivide_s64_signbits(q); // q_sign = q >> 63
__m128i mask = libdivide__u64_to_m128((1ULL << shift) - is_power_of_2);
q = _mm_add_epi64(q, _mm_and_si128(q_sign, mask)); // q = q + (q_sign & mask)
q = libdivide_s64_shift_right_vector(q, shift); // q >>= shift
q = _mm_sub_epi64(_mm_xor_si128(q, sign), sign); // q = (q ^ sign) - sign
return q;
}
#endif
/////////// C++ stuff
#ifdef __cplusplus
// Our divider struct is templated on both a type (like uint64_t) and an
// algorithm index. BRANCHFULL is the default algorithm, BRANCHFREE is the
// branchfree variant, and the indexed variants are for unswitching.
enum {
BRANCHFULL = -1,
BRANCHFREE = -2,
ALGORITHM0 = 0,
ALGORITHM1 = 1,
ALGORITHM2 = 2,
ALGORITHM3 = 3,
ALGORITHM4 = 4
};
namespace libdivide_internal {
#if defined(LIBDIVIDE_USE_SSE2)
#define MAYBE_VECTOR(X) X
#define MAYBE_VECTOR_PARAM(X) __m128i vector_func(__m128i, const X *)
#else
#define MAYBE_VECTOR(X) 0
#define MAYBE_VECTOR_PARAM(X) int unused
#endif
// The following convenience macros are used to build a type of the base
// divider class and give it as template arguments the C functions
// related to the macro name and the macro type paramaters.
#define BRANCHFULL_DIVIDER(INT, TYPE) \
typedef base<INT, \
libdivide_##TYPE##_t, \
libdivide_##TYPE##_gen, \
libdivide_##TYPE##_do, \
MAYBE_VECTOR(libdivide_##TYPE##_do_vector)>
#define BRANCHFREE_DIVIDER(INT, TYPE) \
typedef base<INT, \
libdivide_##TYPE##_branchfree_t, \
libdivide_##TYPE##_branchfree_gen, \
libdivide_##TYPE##_branchfree_do, \
MAYBE_VECTOR(libdivide_##TYPE##_branchfree_do_vector)>
#define ALGORITHM_DIVIDER(INT, TYPE, ALGO) \
typedef base<INT, \
libdivide_##TYPE##_t, \
libdivide_##TYPE##_gen, \
libdivide_##TYPE##_do_##ALGO, \
MAYBE_VECTOR(libdivide_##TYPE##_do_vector_##ALGO)>
#define CRASH_DIVIDER(INT, TYPE) \
typedef base<INT, \
libdivide_##TYPE##_t, \
libdivide_##TYPE##_gen, \
libdivide_##TYPE##_crash, \
MAYBE_VECTOR(libdivide_##TYPE##_crash_vector)>
// Base divider, provides storage for the actual divider.
// @IntType: e.g. uint32_t
// @DenomType: e.g. libdivide_u32_t
// @gen_func(): e.g. libdivide_u32_gen
// @do_func(): e.g. libdivide_u32_do
// @MAYBE_VECTOR_PARAM: e.g. libdivide_u32_do_vector
template<typename IntType,
typename DenomType,
DenomType gen_func(IntType),
IntType do_func(IntType, const DenomType *),
MAYBE_VECTOR_PARAM(DenomType)>
struct base {
// Storage for the actual divider
DenomType denom;
// Constructor that takes a divisor value, and applies the gen function
base(IntType d) : denom(gen_func(d)) { }
// Default constructor to allow uninitialized uses in e.g. arrays
base() {}
// Needed for unswitch
base(const DenomType& d) : denom(d) { }
IntType perform_divide(IntType val) const {
return do_func(val, &denom);
}
#if defined(LIBDIVIDE_USE_SSE2)
__m128i perform_divide_vector(__m128i val) const {
return vector_func(val, &denom);
}
#endif
};
// Functions that will never be called but are required to be able
// to use unswitch in C++ template code. Unsigned has fewer algorithms
// than signed i.e. alg3 and alg4 are not defined for unsigned. In
// order to make templates compile we need to define unsigned alg3 and
// alg4 as crash functions.
uint32_t libdivide_u32_crash(uint32_t, const libdivide_u32_t *) { exit(-1); }
uint64_t libdivide_u64_crash(uint64_t, const libdivide_u64_t *) { exit(-1); }
#if defined(LIBDIVIDE_USE_SSE2)
__m128i libdivide_u32_crash_vector(__m128i, const libdivide_u32_t *) { exit(-1); }
__m128i libdivide_u64_crash_vector(__m128i, const libdivide_u64_t *) { exit(-1); }
#endif
template<typename T, int ALGO> struct dispatcher { };
// Templated dispatch using partial specialization
template<> struct dispatcher<int32_t, BRANCHFULL> { BRANCHFULL_DIVIDER(int32_t, s32) divider; };
template<> struct dispatcher<int32_t, BRANCHFREE> { BRANCHFREE_DIVIDER(int32_t, s32) divider; };
template<> struct dispatcher<int32_t, ALGORITHM0> { ALGORITHM_DIVIDER(int32_t, s32, alg0) divider; };
template<> struct dispatcher<int32_t, ALGORITHM1> { ALGORITHM_DIVIDER(int32_t, s32, alg1) divider; };
template<> struct dispatcher<int32_t, ALGORITHM2> { ALGORITHM_DIVIDER(int32_t, s32, alg2) divider; };
template<> struct dispatcher<int32_t, ALGORITHM3> { ALGORITHM_DIVIDER(int32_t, s32, alg3) divider; };
template<> struct dispatcher<int32_t, ALGORITHM4> { ALGORITHM_DIVIDER(int32_t, s32, alg4) divider; };
template<> struct dispatcher<uint32_t, BRANCHFULL> { BRANCHFULL_DIVIDER(uint32_t, u32) divider; };
template<> struct dispatcher<uint32_t, BRANCHFREE> { BRANCHFREE_DIVIDER(uint32_t, u32) divider; };
template<> struct dispatcher<uint32_t, ALGORITHM0> { ALGORITHM_DIVIDER(uint32_t, u32, alg0) divider; };
template<> struct dispatcher<uint32_t, ALGORITHM1> { ALGORITHM_DIVIDER(uint32_t, u32, alg1) divider; };
template<> struct dispatcher<uint32_t, ALGORITHM2> { ALGORITHM_DIVIDER(uint32_t, u32, alg2) divider; };
template<> struct dispatcher<uint32_t, ALGORITHM3> { CRASH_DIVIDER(uint32_t, u32) divider; };
template<> struct dispatcher<uint32_t, ALGORITHM4> { CRASH_DIVIDER(uint32_t, u32) divider; };
template<> struct dispatcher<int64_t, BRANCHFULL> { BRANCHFULL_DIVIDER(int64_t, s64) divider; };
template<> struct dispatcher<int64_t, BRANCHFREE> { BRANCHFREE_DIVIDER(int64_t, s64) divider; };
template<> struct dispatcher<int64_t, ALGORITHM0> { ALGORITHM_DIVIDER (int64_t, s64, alg0) divider; };
template<> struct dispatcher<int64_t, ALGORITHM1> { ALGORITHM_DIVIDER (int64_t, s64, alg1) divider; };
template<> struct dispatcher<int64_t, ALGORITHM2> { ALGORITHM_DIVIDER (int64_t, s64, alg2) divider; };
template<> struct dispatcher<int64_t, ALGORITHM3> { ALGORITHM_DIVIDER (int64_t, s64, alg3) divider; };
template<> struct dispatcher<int64_t, ALGORITHM4> { ALGORITHM_DIVIDER (int64_t, s64, alg4) divider; };
template<> struct dispatcher<uint64_t, BRANCHFULL> { BRANCHFULL_DIVIDER(uint64_t, u64) divider; };
template<> struct dispatcher<uint64_t, BRANCHFREE> { BRANCHFREE_DIVIDER(uint64_t, u64) divider; };
template<> struct dispatcher<uint64_t, ALGORITHM0> { ALGORITHM_DIVIDER(uint64_t, u64, alg0) divider; };
template<> struct dispatcher<uint64_t, ALGORITHM1> { ALGORITHM_DIVIDER(uint64_t, u64, alg1) divider; };
template<> struct dispatcher<uint64_t, ALGORITHM2> { ALGORITHM_DIVIDER(uint64_t, u64, alg2) divider; };
template<> struct dispatcher<uint64_t, ALGORITHM3> { CRASH_DIVIDER(uint64_t, u64) divider; };
template<> struct dispatcher<uint64_t, ALGORITHM4> { CRASH_DIVIDER(uint64_t, u64) divider; };
// Overloads that don't depend on the algorithm
inline int32_t recover(const libdivide_s32_t *s) { return libdivide_s32_recover(s); }
inline uint32_t recover(const libdivide_u32_t *s) { return libdivide_u32_recover(s); }
inline int64_t recover(const libdivide_s64_t *s) { return libdivide_s64_recover(s); }
inline uint64_t recover(const libdivide_u64_t *s) { return libdivide_u64_recover(s); }
inline int32_t recover(const libdivide_s32_branchfree_t *s) { return libdivide_s32_branchfree_recover(s); }
inline uint32_t recover(const libdivide_u32_branchfree_t *s) { return libdivide_u32_branchfree_recover(s); }
inline int64_t recover(const libdivide_s64_branchfree_t *s) { return libdivide_s64_branchfree_recover(s); }
inline uint64_t recover(const libdivide_u64_branchfree_t *s) { return libdivide_u64_branchfree_recover(s); }
inline int get_algorithm(const libdivide_s32_t *s) { return libdivide_s32_get_algorithm(s); }
inline int get_algorithm(const libdivide_u32_t *s) { return libdivide_u32_get_algorithm(s); }
inline int get_algorithm(const libdivide_s64_t *s) { return libdivide_s64_get_algorithm(s); }
inline int get_algorithm(const libdivide_u64_t *s) { return libdivide_u64_get_algorithm(s); }
// Fallback for branchfree variants, which do not support unswitching
template<typename T> int get_algorithm(const T *) { return -1; }
}
// This is the main divider class for use by the user (C++ API).
// The divider itself is stored in the div variable who's
// type is chosen by the dispatcher based on the template paramaters.
template<typename T, int ALGO = BRANCHFULL>
class divider
{
private:
// Here's the actual divider
typedef typename libdivide_internal::dispatcher<T, ALGO>::divider div_t;
div_t div;
// unswitch() friend declaration
template<int NEW_ALGO, typename S>
friend divider<S, NEW_ALGO> unswitch(const divider<S, BRANCHFULL> & d);
// Constructor used by the unswitch friend
divider(const div_t& denom) : div(denom) { }
public:
// Ordinary constructor that takes the divisor as a parameter
divider(T n) : div(n) { }
// Default constructor. We leave this deliberately undefined so that
// creating an array of divider and then initializing them
// doesn't slow us down.
divider() { }
// Divides the parameter by the divisor, returning the quotient
T perform_divide(T val) const {
return div.perform_divide(val);
}
// Recovers the divisor that was used to initialize the divider
T recover_divisor() const {
return libdivide_internal::recover(&div.denom);
}
#if defined(LIBDIVIDE_USE_SSE2)
// Treats the vector as either two or four packed values (depending on the
// size), and divides each of them by the divisor,
// returning the packed quotients.
__m128i perform_divide_vector(__m128i val) const {
return div.perform_divide_vector(val);
}
#endif
// Returns the index of algorithm, for use in the unswitch function. Does
// not apply to branchfree variant.
// Returns the algorithm for unswitching.
int get_algorithm() const {
return libdivide_internal::get_algorithm(&div.denom);
}
bool operator==(const divider<T, ALGO>& him) const {
return div.denom.magic == him.div.denom.magic &&
div.denom.more == him.div.denom.more;
}
bool operator!=(const divider<T, ALGO>& him) const {
return !(*this == him);
}
};
#if __cplusplus >= 201103L || \
(defined(_MSC_VER) && _MSC_VER >= 1800)
// libdivdie::branchfree_divider<T>
template <typename T>
using branchfree_divider = divider<T, BRANCHFREE>;
#endif
// Returns a divider specialized for the given algorithm
template<int NEW_ALGO, typename T>
divider<T, NEW_ALGO> unswitch(const divider<T, BRANCHFULL>& d) {
return divider<T, NEW_ALGO>(d.div.denom);
}
// Overload of the / operator for scalar division
template<typename int_type, int ALGO>
int_type operator/(int_type numer, const divider<int_type, ALGO>& denom) {
return denom.perform_divide(numer);
}
// Overload of the /= operator for scalar division
template<typename int_type, int ALGO>
int_type operator/=(int_type& numer, const divider<int_type, ALGO>& denom) {
numer = denom.perform_divide(numer);
return numer;
}
#if defined(LIBDIVIDE_USE_SSE2)
// Overload of the / operator for vector division
template<typename int_type, int ALGO>
__m128i operator/(__m128i numer, const divider<int_type, ALGO>& denom) {
return denom.perform_divide_vector(numer);
}
// Overload of the /= operator for vector division
template<typename int_type, int ALGO>
__m128i operator/=(__m128i& numer, const divider<int_type, ALGO>& denom) {
numer = denom.perform_divide_vector(numer);
return numer;
}
#endif
} // namespace libdivide
} // anonymous namespace
#endif // __cplusplus
#endif // LIBDIVIDE_H
#ifndef LOCAL
#pragma GCC optimize ("O3")
#endif
#include <bits/stdc++.h>
using namespace std;
#define sim template < class c
#define ris return * this
#define dor > debug & operator <<
#define eni(x) sim > typename \
enable_if<sizeof dud<c>(0) x 1, debug&>::type operator<<(c i) {
sim > struct rge { c b, e; };
sim > rge<c> range(c i, c j) { return {i, j}; }
sim > auto dud(c* x) -> decltype(cerr << *x, 0);
sim > char dud(...);
struct debug {
#ifdef LOCAL
~debug() { cerr << endl; }
eni(!=) cerr << boolalpha << i; ris; }
eni(==) ris << range(begin(i), end(i)); }
sim, class b dor(pair < b, c > d) {
ris << "(" << d.first << ", " << d.second << ")";
}
sim dor(rge<c> d) {
*this << "[";
for (c it = d.b; it != d.e; ++it)
*this << ", " + 2 * (it == d.b) << *it;
ris << "]";
}
#else
sim dor(const c&) { ris; }
#endif
};
#define imie(x...) " [" #x ": " << (x) << "] "
#include <ext/pb_ds/assoc_container.hpp>
#include <ext/pb_ds/tree_policy.hpp>
template <typename A, typename B>
using unordered_map2 = __gnu_pbds::gp_hash_table<A, B>;
using namespace __gnu_pbds;
template <typename T> using ordered_set =
__gnu_pbds::tree<T, __gnu_pbds::null_type, less<T>, __gnu_pbds::rb_tree_tag,
__gnu_pbds::tree_order_statistics_node_update>;
// ordered_set<int> s; s.insert(1); s.insert(2);
// s.order_of_key(1); // Out: 0.
// *s.find_by_order(1); // Out: 2.
using ld = long double;
using ll = long long;
int mod = 1000 * 1000 * 1000 + 7;
libdivide::libdivide_u64_t fast_mod;
int Moduluj(uint64_t x) {
return x - mod * libdivide::libdivide_u64_do(x, &fast_mod);
}
void OdejmijOd(int& a, int b) { a -= b; if (a < 0) a += mod; }
int Odejmij(int a, int b) { OdejmijOd(a, b); return a; }
void DodajDo(int& a, int b) { a += b; if (a >= mod) a -= mod; }
int Dodaj(int a, int b) { DodajDo(a, b); return a; }
int Mnoz(int a, int b) { return Moduluj(uint64_t(a) * b); }
void MnozDo(int& a, int b) { a = Mnoz(a, b); }
int Pot(int a, ll b) { int res = 1; while (b) { if (b % 2 == 1) MnozDo(res, a); a = Mnoz(a, a); b /= 2; } return res; }
int Odw(int a) { return Pot(a, mod - 2); }
void PodzielDo(int& a, int b) { MnozDo(a, Odw(b)); }
int Podziel(int a, int b) { return Mnoz(a, Odw(b)); }
template <typename T> T Maxi(T& a, T b) { return a = max(a, b); }
template <typename T> T Mini(T& a, T b) { return a = min(a, b); }
constexpr int nax = 1005;
constexpr int kax = 12;
vector<int> Pousuwaj(const vector<int>& v) {
vector<int> res;
const int n = (int) v.size();
for (int i = 0; i < n; i++) {
int ile = 0;
if (i > 0 and v[i - 1] > v[i]) ile++;
if (i + 1 < n and v[i + 1] > v[i]) ile++;
if (ile == 0) {
res.push_back(v[i]);
}
}
return res;
}
int Licz(int ile, const vector<int>& v) {
if (ile == 0) {
if (v.empty()) return 0;
if ((int) v.size() == 1) return 1;
return Licz(0, Pousuwaj(v)) + 1;
} else if (ile == 1) {
assert(!v.empty());
if ((int) v.size() == 1) return 0;
return Licz(1, Pousuwaj(v)) + 1;
} else if (ile == 2) {
assert((int) v.size() >= 2);
if ((int) v.size() == 2) return 0;
return Licz(2, Pousuwaj(v)) + 1;
} else {
assert(false);
}
}
int Brut(int n, int k, int lewo, int prawo) {
const int ile = lewo + prawo;
vector<int> v(n);
iota(v.begin(), v.end(), 0);
int a = 0, b = n;
if (ile == 0) {
// nic.
} else if (ile == 1) {
v.push_back(numeric_limits<int>::max());
} else if (ile == 2) {
v.insert(v.begin(), numeric_limits<int>::max());
v.push_back(numeric_limits<int>::max() - 1);
a++;
b++;
} else {
assert(false);
}
int wyn = 0;
map<int, int> licz;
do {
//if (n == 3 and k == 2) debug() << imie(v);
if (Licz(ile, v) == k) {
if (n == 6 and k == 3 and lewo and !prawo) {
licz[max_element(v.begin() + a, v.begin() + b) - v.begin()]++;
}
wyn++;
}
} while (next_permutation(v.begin() + a, v.begin() + b));
if (n == 6 and k == 3 and lewo and !prawo) {
debug() << imie(n) imie(k) imie(lewo) imie(prawo) imie(licz);
}
return Moduluj(wyn);
}
int C[nax][nax];
int Dp(int n, int k, int lewo, int prawo);
int Pref(int n, int k, int lewo, int prawo);
int Pref_(int n, int k, int lewo, int prawo) {
return Dodaj(Dp(n, k, lewo, prawo), Pref(n, k - 1, lewo, prawo));
}
int Pref(int n, int k, int lewo, int prawo) {
if (n < 0 or k < 0) return 0;
if (lewo > prawo) swap(lewo, prawo);
static int dp[nax][kax][2][2];
static bool juz[nax][kax][2][2];
if (juz[n][k][lewo][prawo]) {
return dp[n][k][lewo][prawo];
}
juz[n][k][lewo][prawo] = true;
return dp[n][k][lewo][prawo] = Pref_(n, k, lewo, prawo);
}
int Dp_(int n, int k, int lewo, int prawo) {
if (n == 0) {
return k == 0;
}
if (n == 1) {
return k == 1;
}
int res = 0;
if (lewo) {
DodajDo(res, Dp(n - 1, k, 1, prawo));
} else {
DodajDo(res, Dp(n - 1, k - 1, 1, prawo));
}
if (prawo) {
DodajDo(res, Dp(n - 1, k, lewo, 1));
} else {
DodajDo(res, Dp(n - 1, k - 1, lewo, 1));
}
for (int i = 2; i < n; i++) {
const int a = i - 1;
const int b = n - i;
int ile = 0;
if (lewo == 0 and prawo == 0) {
DodajDo(ile, Mnoz(Dp(a, k - 1, lewo, 1), Pref(b, k - 1, 1, prawo)));
DodajDo(ile, Mnoz(Pref(a, k - 1, lewo, 1), Dp(b, k - 1, 1, prawo)));
OdejmijOd(ile, Mnoz(Dp(a, k - 1, lewo, 1), Dp(b, k - 1, 1, prawo)));
} else if (lewo == 1 and prawo == 1) {
DodajDo(ile, Mnoz(Dp(a, k - 1, lewo, 1), Dp(b, k - 1, 1, prawo)));
DodajDo(ile, Mnoz(Dp(a, k, lewo, 1), Pref(b, k - 1, 1, prawo)));
DodajDo(ile, Mnoz(Pref(a, k - 1, lewo, 1), Dp(b, k, 1, prawo)));
} else if (lewo == 1) {
DodajDo(ile, Mnoz(Dp(a, k - 1, lewo, 1), Pref(b, k, 1, prawo)));
DodajDo(ile, Mnoz(Pref(a, k - 2, lewo, 1), Dp(b, k, 1, prawo)));
} else if (prawo == 1) {
DodajDo(ile, Mnoz(Pref(a, k, lewo, 1), Dp(b, k - 1, 1, prawo)));
DodajDo(ile, Mnoz(Dp(a, k, lewo, 1), Pref(b, k - 2, 1, prawo)));
} else {
assert(false);
}
// DodajDo(ile, Mnoz(Dp(a, k - 1, lewo, 1), Pref(b, k - 1, 1, prawo)));
// DodajDo(ile, Mnoz(Pref(a, k - 1, lewo, 1), Dp(b, k - 1, 1, prawo)));
// OdejmijOd(ile, Mnoz(Dp(a, k - 1, lewo, 1), Dp(b, k - 1, 1, prawo)));
// debug() << imie(a) imie(b) imie(lewo) imie(prawo);
// debug() << imie(Dp(a, k - 1, lewo, 1)) imie(Pref(a, k - 1, lewo, 1));
// debug() << imie(Dp(b, k - 1, 1, prawo)) imie(Pref(b, k - 1, 1, prawo));
// debug() << imie(ile) imie(C[a + b][a]);
DodajDo(res, Mnoz(ile, C[a + b][a]));
}
debug() << "Dp(" imie(n) imie(k) imie(lewo) imie(prawo) ") = " << res;
#ifdef LOCAL
// const int b = Brut(n, k, lewo, prawo);
// if (b != res) {
// debug() << imie(res) imie(b);
// assert(false);
// }
#endif
return res;
}
int Dp(int n, int k, int lewo, int prawo) {
if (n < 0 or k < 0) return 0;
if (lewo > prawo) swap(lewo, prawo);
static int tab[nax][kax][2][2];
static bool juz[nax][kax][2][2];
if (juz[n][k][lewo][prawo]) {
return tab[n][k][lewo][prawo];
}
juz[n][k][lewo][prawo] = true;
return tab[n][k][lewo][prawo] = Dp_(n, k, lewo, prawo);
}
int Daj(int n, int k) {
if (!(1 <= n and n < nax and 1 <= k and k + 1 < kax)) return 0;
const int res = Dp(n, k + 1, 0, 0);
assert(res >= 0);
return res;
}
int main() {
ios_base::sync_with_stdio(0);
cin.tie(0);
int n, k;
cin >> n >> k >> mod;
fast_mod = libdivide::libdivide_u64_gen(mod);
for (int A = 0; A < nax; A++) {
C[A][0] = C[A][A] = 1;
for (int B = 1; B < A; B++) {
C[A][B] = Dodaj(C[A - 1][B - 1], C[A - 1][B]);
}
}
cout << Daj(n, k) << endl;
// while (cin >> n >> k) {
// cout << Daj(n, k) << endl;
// }
// int suma = 0;
// for (n = 1; n <= 1000; n++) {
// for (k = 1; k <= n; k++) {
// DodajDo(suma, Mnoz(Mnoz(n, k), Daj(n, k)));
// }
// }
// cout << suma << endl;
// // wynik: 59180009
// for (int n = 0; n <= 10; n++) {
// for (int k = 0; k <= n; k++) {
// Dp(n, k, 0, 0);
// }
// }
// printf("[n = %2d]:", n_);
// for (int i = 0; i <= n_; i++) {
// printf(" %7d", Dp(n_, i, 0, 0));
// }
// printf("\n");
return 0;
}
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2222 2223 2224 2225 2226 2227 2228 2229 2230 2231 2232 2233 2234 2235 2236 2237 2238 2239 2240 2241 2242 2243 2244 2245 2246 2247 2248 2249 2250 2251 2252 2253 2254 2255 2256 2257 2258 2259 2260 2261 2262 2263 2264 2265 2266 2267 2268 2269 2270 2271 2272 2273 2274 2275 2276 2277 2278 2279 2280 2281 2282 2283 2284 2285 2286 2287 2288 2289 2290 2291 2292 2293 2294 2295 2296 2297 2298 2299 2300 2301 2302 2303 2304 2305 2306 2307 2308 2309 2310 2311 2312 2313 2314 2315 2316 2317 2318 2319 2320 2321 2322 2323 2324 2325 2326 2327 2328 2329 2330 2331 2332 2333 2334 2335 2336 2337 2338 2339 | // libdivide.h // Copyright 2010 - 2018 ridiculous_fish // // libdivide is dual-licensed under the Boost or zlib licenses. // You may use libdivide under the terms of either of these. // See LICENSE.txt for more details. #ifndef LIBDIVIDE_H #define LIBDIVIDE_H #if defined(_MSC_VER) // disable warning C4146: unary minus operator applied to // unsigned type, result still unsigned #pragma warning(disable: 4146) #define LIBDIVIDE_VC #endif #ifdef __cplusplus #include <cstdlib> #include <cstdio> #else #include <stdlib.h> #include <stdio.h> #endif #include <stdint.h> #if defined(LIBDIVIDE_USE_SSE2) #include <emmintrin.h> #endif #if defined(LIBDIVIDE_VC) #include <intrin.h> #endif #ifndef __has_builtin #define __has_builtin(x) 0 // Compatibility with non-clang compilers. #endif #if defined(__SIZEOF_INT128__) #define HAS_INT128_T #endif #if defined(__x86_64__) || defined(_WIN64) || defined(_M_X64) #define LIBDIVIDE_IS_X86_64 #endif #if defined(__i386__) #define LIBDIVIDE_IS_i386 #endif #if defined(__GNUC__) || defined(__clang__) #define LIBDIVIDE_GCC_STYLE_ASM #endif #if defined(__cplusplus) || defined(LIBDIVIDE_VC) #define LIBDIVIDE_FUNCTION __FUNCTION__ #else #define LIBDIVIDE_FUNCTION __func__ #endif #define LIBDIVIDE_ERROR(msg) \ do { \ fprintf(stderr, "libdivide.h:%d: %s(): Error: %s\n", \ __LINE__, LIBDIVIDE_FUNCTION, msg); \ exit(-1); \ } while (0) #if defined(LIBDIVIDE_ASSERTIONS_ON) #define LIBDIVIDE_ASSERT(x) \ do { \ if (!(x)) { \ fprintf(stderr, "libdivide.h:%d: %s(): Assertion failed: %s\n", \ __LINE__, LIBDIVIDE_FUNCTION, #x); \ exit(-1); \ } \ } while (0) #else #define LIBDIVIDE_ASSERT(x) #endif // libdivide may use the pmuldq (vector signed 32x32->64 mult instruction) // which is in SSE 4.1. However, signed multiplication can be emulated // efficiently with unsigned multiplication, and SSE 4.1 is currently rare, so // it is OK to not turn this on. #ifdef LIBDIVIDE_USE_SSE4_1 #include <smmintrin.h> #endif #ifdef __cplusplus // We place libdivide within the libdivide namespace, and that goes in an // anonymous namespace so that the functions are only visible to files that // #include this header and don't get external linkage. At least that's the // theory. namespace { namespace libdivide { #endif // Explanation of "more" field: bit 6 is whether to use shift path. If we are // using the shift path, bit 7 is whether the divisor is negative in the signed // case; in the unsigned case it is 0. Bits 0-4 is shift value (for shift // path or mult path). In 32 bit case, bit 5 is always 0. We use bit 7 as the // "negative divisor indicator" so that we can use sign extension to // efficiently go to a full-width -1. // // u32: [0-4] shift value // [5] ignored // [6] add indicator // [7] shift path // // s32: [0-4] shift value // [5] shift path // [6] add indicator // [7] indicates negative divisor // // u64: [0-5] shift value // [6] add indicator // [7] shift path // // s64: [0-5] shift value // [6] add indicator // [7] indicates negative divisor // magic number of 0 indicates shift path (we ran out of bits!) // // In s32 and s64 branchfree modes, the magic number is negated according to // whether the divisor is negated. In branchfree strategy, it is not negated. enum { LIBDIVIDE_32_SHIFT_MASK = 0x1F, LIBDIVIDE_64_SHIFT_MASK = 0x3F, LIBDIVIDE_ADD_MARKER = 0x40, LIBDIVIDE_U32_SHIFT_PATH = 0x80, LIBDIVIDE_U64_SHIFT_PATH = 0x80, LIBDIVIDE_S32_SHIFT_PATH = 0x20, LIBDIVIDE_NEGATIVE_DIVISOR = 0x80 }; // pack divider structs to prevent compilers from padding. // This reduces memory usage by up to 43% when using a large // array of libdivide dividers and improves performance // by up to 10% because of reduced memory bandwidth. #pragma pack(push, 1) struct libdivide_u32_t { uint32_t magic; uint8_t more; }; struct libdivide_s32_t { int32_t magic; uint8_t more; }; struct libdivide_u64_t { uint64_t magic; uint8_t more; }; struct libdivide_s64_t { int64_t magic; uint8_t more; }; struct libdivide_u32_branchfree_t { uint32_t magic; uint8_t more; }; struct libdivide_s32_branchfree_t { int32_t magic; uint8_t more; }; struct libdivide_u64_branchfree_t { uint64_t magic; uint8_t more; }; struct libdivide_s64_branchfree_t { int64_t magic; uint8_t more; }; #pragma pack(pop) #ifndef LIBDIVIDE_API #ifdef __cplusplus // In C++, we don't want our public functions to be static, because // they are arguments to templates and static functions can't do that. // They get internal linkage through virtue of the anonymous namespace. // In C, they should be static. #define LIBDIVIDE_API #else #define LIBDIVIDE_API static inline #endif #endif LIBDIVIDE_API struct libdivide_s32_t libdivide_s32_gen(int32_t y); LIBDIVIDE_API struct libdivide_u32_t libdivide_u32_gen(uint32_t y); LIBDIVIDE_API struct libdivide_s64_t libdivide_s64_gen(int64_t y); LIBDIVIDE_API struct libdivide_u64_t libdivide_u64_gen(uint64_t y); LIBDIVIDE_API struct libdivide_s32_branchfree_t libdivide_s32_branchfree_gen(int32_t y); LIBDIVIDE_API struct libdivide_u32_branchfree_t libdivide_u32_branchfree_gen(uint32_t y); LIBDIVIDE_API struct libdivide_s64_branchfree_t libdivide_s64_branchfree_gen(int64_t y); LIBDIVIDE_API struct libdivide_u64_branchfree_t libdivide_u64_branchfree_gen(uint64_t y); LIBDIVIDE_API int32_t libdivide_s32_do(int32_t numer, const struct libdivide_s32_t *denom); LIBDIVIDE_API uint32_t libdivide_u32_do(uint32_t numer, const struct libdivide_u32_t *denom); LIBDIVIDE_API int64_t libdivide_s64_do(int64_t numer, const struct libdivide_s64_t *denom); LIBDIVIDE_API uint64_t libdivide_u64_do(uint64_t y, const struct libdivide_u64_t *denom); LIBDIVIDE_API int32_t libdivide_s32_branchfree_do(int32_t numer, const struct libdivide_s32_branchfree_t *denom); LIBDIVIDE_API uint32_t libdivide_u32_branchfree_do(uint32_t numer, const struct libdivide_u32_branchfree_t *denom); LIBDIVIDE_API int64_t libdivide_s64_branchfree_do(int64_t numer, const struct libdivide_s64_branchfree_t *denom); LIBDIVIDE_API uint64_t libdivide_u64_branchfree_do(uint64_t y, const struct libdivide_u64_branchfree_t *denom); LIBDIVIDE_API int32_t libdivide_s32_recover(const struct libdivide_s32_t *denom); LIBDIVIDE_API uint32_t libdivide_u32_recover(const struct libdivide_u32_t *denom); LIBDIVIDE_API int64_t libdivide_s64_recover(const struct libdivide_s64_t *denom); LIBDIVIDE_API uint64_t libdivide_u64_recover(const struct libdivide_u64_t *denom); LIBDIVIDE_API int32_t libdivide_s32_branchfree_recover(const struct libdivide_s32_branchfree_t *denom); LIBDIVIDE_API uint32_t libdivide_u32_branchfree_recover(const struct libdivide_u32_branchfree_t *denom); LIBDIVIDE_API int64_t libdivide_s64_branchfree_recover(const struct libdivide_s64_branchfree_t *denom); LIBDIVIDE_API uint64_t libdivide_u64_branchfree_recover(const struct libdivide_u64_branchfree_t *denom); LIBDIVIDE_API int libdivide_u32_get_algorithm(const struct libdivide_u32_t *denom); LIBDIVIDE_API uint32_t libdivide_u32_do_alg0(uint32_t numer, const struct libdivide_u32_t *denom); LIBDIVIDE_API uint32_t libdivide_u32_do_alg1(uint32_t numer, const struct libdivide_u32_t *denom); LIBDIVIDE_API uint32_t libdivide_u32_do_alg2(uint32_t numer, const struct libdivide_u32_t *denom); LIBDIVIDE_API int libdivide_u64_get_algorithm(const struct libdivide_u64_t *denom); LIBDIVIDE_API uint64_t libdivide_u64_do_alg0(uint64_t numer, const struct libdivide_u64_t *denom); LIBDIVIDE_API uint64_t libdivide_u64_do_alg1(uint64_t numer, const struct libdivide_u64_t *denom); LIBDIVIDE_API uint64_t libdivide_u64_do_alg2(uint64_t numer, const struct libdivide_u64_t *denom); LIBDIVIDE_API int libdivide_s32_get_algorithm(const struct libdivide_s32_t *denom); LIBDIVIDE_API int32_t libdivide_s32_do_alg0(int32_t numer, const struct libdivide_s32_t *denom); LIBDIVIDE_API int32_t libdivide_s32_do_alg1(int32_t numer, const struct libdivide_s32_t *denom); LIBDIVIDE_API int32_t libdivide_s32_do_alg2(int32_t numer, const struct libdivide_s32_t *denom); LIBDIVIDE_API int32_t libdivide_s32_do_alg3(int32_t numer, const struct libdivide_s32_t *denom); LIBDIVIDE_API int32_t libdivide_s32_do_alg4(int32_t numer, const struct libdivide_s32_t *denom); LIBDIVIDE_API int libdivide_s64_get_algorithm(const struct libdivide_s64_t *denom); LIBDIVIDE_API int64_t libdivide_s64_do_alg0(int64_t numer, const struct libdivide_s64_t *denom); LIBDIVIDE_API int64_t libdivide_s64_do_alg1(int64_t numer, const struct libdivide_s64_t *denom); LIBDIVIDE_API int64_t libdivide_s64_do_alg2(int64_t numer, const struct libdivide_s64_t *denom); LIBDIVIDE_API int64_t libdivide_s64_do_alg3(int64_t numer, const struct libdivide_s64_t *denom); LIBDIVIDE_API int64_t libdivide_s64_do_alg4(int64_t numer, const struct libdivide_s64_t *denom); #if defined(LIBDIVIDE_USE_SSE2) LIBDIVIDE_API __m128i libdivide_u32_do_vector(__m128i numers, const struct libdivide_u32_t *denom); LIBDIVIDE_API __m128i libdivide_s32_do_vector(__m128i numers, const struct libdivide_s32_t *denom); LIBDIVIDE_API __m128i libdivide_u64_do_vector(__m128i numers, const struct libdivide_u64_t *denom); LIBDIVIDE_API __m128i libdivide_s64_do_vector(__m128i numers, const struct libdivide_s64_t *denom); LIBDIVIDE_API __m128i libdivide_u32_do_vector_alg0(__m128i numers, const struct libdivide_u32_t *denom); LIBDIVIDE_API __m128i libdivide_u32_do_vector_alg1(__m128i numers, const struct libdivide_u32_t *denom); LIBDIVIDE_API __m128i libdivide_u32_do_vector_alg2(__m128i numers, const struct libdivide_u32_t *denom); LIBDIVIDE_API __m128i libdivide_s32_do_vector_alg0(__m128i numers, const struct libdivide_s32_t *denom); LIBDIVIDE_API __m128i libdivide_s32_do_vector_alg1(__m128i numers, const struct libdivide_s32_t *denom); LIBDIVIDE_API __m128i libdivide_s32_do_vector_alg2(__m128i numers, const struct libdivide_s32_t *denom); LIBDIVIDE_API __m128i libdivide_s32_do_vector_alg3(__m128i numers, const struct libdivide_s32_t *denom); LIBDIVIDE_API __m128i libdivide_s32_do_vector_alg4(__m128i numers, const struct libdivide_s32_t *denom); LIBDIVIDE_API __m128i libdivide_u64_do_vector_alg0(__m128i numers, const struct libdivide_u64_t *denom); LIBDIVIDE_API __m128i libdivide_u64_do_vector_alg1(__m128i numers, const struct libdivide_u64_t *denom); LIBDIVIDE_API __m128i libdivide_u64_do_vector_alg2(__m128i numers, const struct libdivide_u64_t *denom); LIBDIVIDE_API __m128i libdivide_s64_do_vector_alg0(__m128i numers, const struct libdivide_s64_t *denom); LIBDIVIDE_API __m128i libdivide_s64_do_vector_alg1(__m128i numers, const struct libdivide_s64_t *denom); LIBDIVIDE_API __m128i libdivide_s64_do_vector_alg2(__m128i numers, const struct libdivide_s64_t *denom); LIBDIVIDE_API __m128i libdivide_s64_do_vector_alg3(__m128i numers, const struct libdivide_s64_t *denom); LIBDIVIDE_API __m128i libdivide_s64_do_vector_alg4(__m128i numers, const struct libdivide_s64_t *denom); LIBDIVIDE_API __m128i libdivide_u32_branchfree_do_vector(__m128i numers, const struct libdivide_u32_branchfree_t *denom); LIBDIVIDE_API __m128i libdivide_s32_branchfree_do_vector(__m128i numers, const struct libdivide_s32_branchfree_t *denom); LIBDIVIDE_API __m128i libdivide_u64_branchfree_do_vector(__m128i numers, const struct libdivide_u64_branchfree_t *denom); LIBDIVIDE_API __m128i libdivide_s64_branchfree_do_vector(__m128i numers, const struct libdivide_s64_branchfree_t *denom); #endif //////// Internal Utility Functions static inline uint32_t libdivide__mullhi_u32(uint32_t x, uint32_t y) { uint64_t xl = x, yl = y; uint64_t rl = xl * yl; return (uint32_t)(rl >> 32); } static uint64_t libdivide__mullhi_u64(uint64_t x, uint64_t y) { #if defined(LIBDIVIDE_VC) && defined(LIBDIVIDE_IS_X86_64) return __umulh(x, y); #elif defined(HAS_INT128_T) __uint128_t xl = x, yl = y; __uint128_t rl = xl * yl; return (uint64_t)(rl >> 64); #else // full 128 bits are x0 * y0 + (x0 * y1 << 32) + (x1 * y0 << 32) + (x1 * y1 << 64) uint32_t mask = 0xFFFFFFFF; uint32_t x0 = (uint32_t)(x & mask); uint32_t x1 = (uint32_t)(x >> 32); uint32_t y0 = (uint32_t)(y & mask); uint32_t y1 = (uint32_t)(y >> 32); uint32_t x0y0_hi = libdivide__mullhi_u32(x0, y0); uint64_t x0y1 = x0 * (uint64_t)y1; uint64_t x1y0 = x1 * (uint64_t)y0; uint64_t x1y1 = x1 * (uint64_t)y1; uint64_t temp = x1y0 + x0y0_hi; uint64_t temp_lo = temp & mask; uint64_t temp_hi = temp >> 32; return x1y1 + temp_hi + ((temp_lo + x0y1) >> 32); #endif } static inline int64_t libdivide__mullhi_s64(int64_t x, int64_t y) { #if defined(LIBDIVIDE_VC) && defined(LIBDIVIDE_IS_X86_64) return __mulh(x, y); #elif defined(HAS_INT128_T) __int128_t xl = x, yl = y; __int128_t rl = xl * yl; return (int64_t)(rl >> 64); #else // full 128 bits are x0 * y0 + (x0 * y1 << 32) + (x1 * y0 << 32) + (x1 * y1 << 64) uint32_t mask = 0xFFFFFFFF; uint32_t x0 = (uint32_t)(x & mask); uint32_t y0 = (uint32_t)(y & mask); int32_t x1 = (int32_t)(x >> 32); int32_t y1 = (int32_t)(y >> 32); uint32_t x0y0_hi = libdivide__mullhi_u32(x0, y0); int64_t t = x1 * (int64_t)y0 + x0y0_hi; int64_t w1 = x0 * (int64_t)y1 + (t & mask); return x1 * (int64_t)y1 + (t >> 32) + (w1 >> 32); #endif } #if defined(LIBDIVIDE_USE_SSE2) static inline __m128i libdivide__u64_to_m128(uint64_t x) { #if defined(LIBDIVIDE_VC) && !defined(_WIN64) // 64 bit windows doesn't seem to have an implementation of any of these // load intrinsics, and 32 bit Visual C++ crashes _declspec(align(16)) uint64_t temp[2] = {x, x}; return _mm_load_si128((const __m128i*)temp); #else // everyone else gets it right return _mm_set1_epi64x(x); #endif } static inline __m128i libdivide_get_FFFFFFFF00000000(void) { // returns the same as _mm_set1_epi64(0xFFFFFFFF00000000ULL) // without touching memory. // optimizes to pcmpeqd on OS X __m128i result = _mm_set1_epi8(-1); return _mm_slli_epi64(result, 32); } static inline __m128i libdivide_get_00000000FFFFFFFF(void) { // returns the same as _mm_set1_epi64(0x00000000FFFFFFFFULL) // without touching memory. // optimizes to pcmpeqd on OS X __m128i result = _mm_set1_epi8(-1); result = _mm_srli_epi64(result, 32); return result; } static inline __m128i libdivide_s64_signbits(__m128i v) { // we want to compute v >> 63, that is, _mm_srai_epi64(v, 63). But there // is no 64 bit shift right arithmetic instruction in SSE2. So we have to // fake it by first duplicating the high 32 bit values, and then using a 32 // bit shift. Another option would be to use _mm_srli_epi64(v, 63) and // then subtract that from 0, but that approach appears to be substantially // slower for unknown reasons __m128i hiBitsDuped = _mm_shuffle_epi32(v, _MM_SHUFFLE(3, 3, 1, 1)); __m128i signBits = _mm_srai_epi32(hiBitsDuped, 31); return signBits; } // Returns an __m128i whose low 32 bits are equal to amt and has zero elsewhere. static inline __m128i libdivide_u32_to_m128i(uint32_t amt) { return _mm_set_epi32(0, 0, 0, amt); } static inline __m128i libdivide_s64_shift_right_vector(__m128i v, int amt) { // implementation of _mm_sra_epi64. Here we have two 64 bit values which // are shifted right to logically become (64 - amt) values, and are then // sign extended from a (64 - amt) bit number. const int b = 64 - amt; __m128i m = libdivide__u64_to_m128(1ULL << (b - 1)); __m128i x = _mm_srl_epi64(v, libdivide_u32_to_m128i(amt)); __m128i result = _mm_sub_epi64(_mm_xor_si128(x, m), m); // result = x^m - m return result; } // Here, b is assumed to contain one 32 bit value repeated four times. // If it did not, the function would not work. static inline __m128i libdivide__mullhi_u32_flat_vector(__m128i a, __m128i b) { __m128i hi_product_0Z2Z = _mm_srli_epi64(_mm_mul_epu32(a, b), 32); __m128i a1X3X = _mm_srli_epi64(a, 32); __m128i mask = libdivide_get_FFFFFFFF00000000(); __m128i hi_product_Z1Z3 = _mm_and_si128(_mm_mul_epu32(a1X3X, b), mask); return _mm_or_si128(hi_product_0Z2Z, hi_product_Z1Z3); // = hi_product_0123 } // Here, y is assumed to contain one 64 bit value repeated twice. static inline __m128i libdivide_mullhi_u64_flat_vector(__m128i x, __m128i y) { // full 128 bits are x0 * y0 + (x0 * y1 << 32) + (x1 * y0 << 32) + (x1 * y1 << 64) __m128i mask = libdivide_get_00000000FFFFFFFF(); // x0 is low half of 2 64 bit values, x1 is high half in low slots __m128i x0 = _mm_and_si128(x, mask); __m128i x1 = _mm_srli_epi64(x, 32); __m128i y0 = _mm_and_si128(y, mask); __m128i y1 = _mm_srli_epi64(y, 32); // x0 happens to have the low half of the two 64 bit values in 32 bit slots // 0 and 2, so _mm_mul_epu32 computes their full product, and then we shift // right by 32 to get just the high values __m128i x0y0_hi = _mm_srli_epi64(_mm_mul_epu32(x0, y0), 32); __m128i x0y1 = _mm_mul_epu32(x0, y1); __m128i x1y0 = _mm_mul_epu32(x1, y0); __m128i x1y1 = _mm_mul_epu32(x1, y1); __m128i temp = _mm_add_epi64(x1y0, x0y0_hi); __m128i temp_lo = _mm_and_si128(temp, mask); __m128i temp_hi = _mm_srli_epi64(temp, 32); temp_lo = _mm_srli_epi64(_mm_add_epi64(temp_lo, x0y1), 32); temp_hi = _mm_add_epi64(x1y1, temp_hi); return _mm_add_epi64(temp_lo, temp_hi); } // y is one 64 bit value repeated twice static inline __m128i libdivide_mullhi_s64_flat_vector(__m128i x, __m128i y) { __m128i p = libdivide_mullhi_u64_flat_vector(x, y); __m128i t1 = _mm_and_si128(libdivide_s64_signbits(x), y); p = _mm_sub_epi64(p, t1); __m128i t2 = _mm_and_si128(libdivide_s64_signbits(y), x); p = _mm_sub_epi64(p, t2); return p; } #ifdef LIBDIVIDE_USE_SSE4_1 // b is one 32 bit value repeated four times. static inline __m128i libdivide_mullhi_s32_flat_vector(__m128i a, __m128i b) { __m128i hi_product_0Z2Z = _mm_srli_epi64(_mm_mul_epi32(a, b), 32); __m128i a1X3X = _mm_srli_epi64(a, 32); __m128i mask = libdivide_get_FFFFFFFF00000000(); __m128i hi_product_Z1Z3 = _mm_and_si128(_mm_mul_epi32(a1X3X, b), mask); return _mm_or_si128(hi_product_0Z2Z, hi_product_Z1Z3); // = hi_product_0123 } #else // SSE2 does not have a signed multiplication instruction, but we can convert // unsigned to signed pretty efficiently. Again, b is just a 32 bit value // repeated four times. static inline __m128i libdivide_mullhi_s32_flat_vector(__m128i a, __m128i b) { __m128i p = libdivide__mullhi_u32_flat_vector(a, b); __m128i t1 = _mm_and_si128(_mm_srai_epi32(a, 31), b); // t1 = (a >> 31) & y, arithmetic shift __m128i t2 = _mm_and_si128(_mm_srai_epi32(b, 31), a); p = _mm_sub_epi32(p, t1); p = _mm_sub_epi32(p, t2); return p; } #endif // LIBDIVIDE_USE_SSE4_1 #endif // LIBDIVIDE_USE_SSE2 static inline int32_t libdivide__count_leading_zeros32(uint32_t val) { #if defined(__GNUC__) || __has_builtin(__builtin_clz) // Fast way to count leading zeros return __builtin_clz(val); #elif defined(LIBDIVIDE_VC) unsigned long result; if (_BitScanReverse(&result, val)) { return 31 - result; } return 0; #else int32_t result = 0; uint32_t hi = 1U << 31; while (~val & hi) { hi >>= 1; result++; } return result; #endif } static inline int32_t libdivide__count_leading_zeros64(uint64_t val) { #if defined(__GNUC__) || __has_builtin(__builtin_clzll) // Fast way to count leading zeros return __builtin_clzll(val); #elif defined(LIBDIVIDE_VC) && defined(_WIN64) unsigned long result; if (_BitScanReverse64(&result, val)) { return 63 - result; } return 0; #else uint32_t hi = val >> 32; uint32_t lo = val & 0xFFFFFFFF; if (hi != 0) return libdivide__count_leading_zeros32(hi); return 32 + libdivide__count_leading_zeros32(lo); #endif } #if (defined(LIBDIVIDE_IS_i386) || defined(LIBDIVIDE_IS_X86_64)) && \ defined(LIBDIVIDE_GCC_STYLE_ASM) // libdivide_64_div_32_to_32: divides a 64 bit uint {u1, u0} by a 32 bit // uint {v}. The result must fit in 32 bits. // Returns the quotient directly and the remainder in *r static uint32_t libdivide_64_div_32_to_32(uint32_t u1, uint32_t u0, uint32_t v, uint32_t *r) { uint32_t result; __asm__("divl %[v]" : "=a"(result), "=d"(*r) : [v] "r"(v), "a"(u0), "d"(u1) ); return result; } #else static uint32_t libdivide_64_div_32_to_32(uint32_t u1, uint32_t u0, uint32_t v, uint32_t *r) { uint64_t n = (((uint64_t)u1) << 32) | u0; uint32_t result = (uint32_t)(n / v); *r = (uint32_t)(n - result * (uint64_t)v); return result; } #endif #if defined(LIBDIVIDE_IS_X86_64) && \ defined(LIBDIVIDE_GCC_STYLE_ASM) static uint64_t libdivide_128_div_64_to_64(uint64_t u1, uint64_t u0, uint64_t v, uint64_t *r) { // u0 -> rax // u1 -> rdx // divq uint64_t result; __asm__("divq %[v]" : "=a"(result), "=d"(*r) : [v] "r"(v), "a"(u0), "d"(u1) ); return result; } #else // Code taken from Hacker's Delight: // http://www.hackersdelight.org/HDcode/divlu.c. // License permits inclusion here per: // http://www.hackersdelight.org/permissions.htm static uint64_t libdivide_128_div_64_to_64(uint64_t u1, uint64_t u0, uint64_t v, uint64_t *r) { const uint64_t b = (1ULL << 32); // Number base (16 bits) uint64_t un1, un0; // Norm. dividend LSD's uint64_t vn1, vn0; // Norm. divisor digits uint64_t q1, q0; // Quotient digits uint64_t un64, un21, un10; // Dividend digit pairs uint64_t rhat; // A remainder int32_t s; // Shift amount for norm // If overflow, set rem. to an impossible value, // and return the largest possible quotient if (u1 >= v) { if (r != NULL) *r = (uint64_t) -1; return (uint64_t) -1; } // count leading zeros s = libdivide__count_leading_zeros64(v); if (s > 0) { // Normalize divisor v = v << s; un64 = (u1 << s) | ((u0 >> (64 - s)) & (-s >> 31)); un10 = u0 << s; // Shift dividend left } else { // Avoid undefined behavior un64 = u1 | u0; un10 = u0; } // Break divisor up into two 32-bit digits vn1 = v >> 32; vn0 = v & 0xFFFFFFFF; // Break right half of dividend into two digits un1 = un10 >> 32; un0 = un10 & 0xFFFFFFFF; // Compute the first quotient digit, q1 q1 = un64 / vn1; rhat = un64 - q1 * vn1; while (q1 >= b || q1 * vn0 > b * rhat + un1) { q1 = q1 - 1; rhat = rhat + vn1; if (rhat >= b) break; } // Multiply and subtract un21 = un64 * b + un1 - q1 * v; // Compute the second quotient digit q0 = un21 / vn1; rhat = un21 - q0 * vn1; while (q0 >= b || q0 * vn0 > b * rhat + un0) { q0 = q0 - 1; rhat = rhat + vn1; if (rhat >= b) break; } // If remainder is wanted, return it if (r != NULL) *r = (un21 * b + un0 - q0 * v) >> s; return q1 * b + q0; } #endif // Bitshift a u128 in place, left (signed_shift > 0) or right (signed_shift < 0) static inline void libdivide_u128_shift(uint64_t *u1, uint64_t *u0, int32_t signed_shift) { if (signed_shift > 0) { uint32_t shift = signed_shift; *u1 <<= shift; *u1 |= *u0 >> (64 - shift); *u0 <<= shift; } else { uint32_t shift = -signed_shift; *u0 >>= shift; *u0 |= *u1 << (64 - shift); *u1 >>= shift; } } // Computes a 128 / 128 -> 64 bit division, with a 128 bit remainder. static uint64_t libdivide_128_div_128_to_64(uint64_t u_hi, uint64_t u_lo, uint64_t v_hi, uint64_t v_lo, uint64_t *r_hi, uint64_t *r_lo) { #if defined(HAS_INT128_T) __uint128_t ufull = u_hi; __uint128_t vfull = v_hi; ufull = (ufull << 64) | u_lo; vfull = (vfull << 64) | v_lo; uint64_t res = (uint64_t)(ufull / vfull); __uint128_t remainder = ufull - (vfull * res); *r_lo = (uint64_t)remainder; *r_hi = (uint64_t)(remainder >> 64); return res; #else // Adapted from "Unsigned Doubleword Division" in Hacker's Delight // We want to compute u / v typedef struct { uint64_t hi; uint64_t lo; } u128_t; u128_t u = {u_hi, u_lo}; u128_t v = {v_hi, v_lo}; if (v.hi == 0) { // divisor v is a 64 bit value, so we just need one 128/64 division // Note that we are simpler than Hacker's Delight here, because we know // the quotient fits in 64 bits whereas Hacker's Delight demands a full // 128 bit quotient *r_hi = 0; return libdivide_128_div_64_to_64(u.hi, u.lo, v.lo, r_lo); } // Here v >= 2**64 // We know that v.hi != 0, so count leading zeros is OK // We have 0 <= n <= 63 uint32_t n = libdivide__count_leading_zeros64(v.hi); // Normalize the divisor so its MSB is 1 u128_t v1t = v; libdivide_u128_shift(&v1t.hi, &v1t.lo, n); uint64_t v1 = v1t.hi; // i.e. v1 = v1t >> 64 // To ensure no overflow u128_t u1 = u; libdivide_u128_shift(&u1.hi, &u1.lo, -1); // Get quotient from divide unsigned insn. uint64_t rem_ignored; uint64_t q1 = libdivide_128_div_64_to_64(u1.hi, u1.lo, v1, &rem_ignored); // Undo normalization and division of u by 2. u128_t q0 = {0, q1}; libdivide_u128_shift(&q0.hi, &q0.lo, n); libdivide_u128_shift(&q0.hi, &q0.lo, -63); // Make q0 correct or too small by 1 // Equivalent to `if (q0 != 0) q0 = q0 - 1;` if (q0.hi != 0 || q0.lo != 0) { q0.hi -= (q0.lo == 0); // borrow q0.lo -= 1; } // Now q0 is correct. // Compute q0 * v as q0v // = (q0.hi << 64 + q0.lo) * (v.hi << 64 + v.lo) // = (q0.hi * v.hi << 128) + (q0.hi * v.lo << 64) + // (q0.lo * v.hi << 64) + q0.lo * v.lo) // Each term is 128 bit // High half of full product (upper 128 bits!) are dropped u128_t q0v = {0, 0}; q0v.hi = q0.hi*v.lo + q0.lo*v.hi + libdivide__mullhi_u64(q0.lo, v.lo); q0v.lo = q0.lo*v.lo; // Compute u - q0v as u_q0v // This is the remainder u128_t u_q0v = u; u_q0v.hi -= q0v.hi + (u.lo < q0v.lo); // second term is borrow u_q0v.lo -= q0v.lo; // Check if u_q0v >= v // This checks if our remainder is larger than the divisor if ((u_q0v.hi > v.hi) || (u_q0v.hi == v.hi && u_q0v.lo >= v.lo)) { // Increment q0 q0.lo += 1; q0.hi += (q0.lo == 0); // carry // Subtract v from remainder u_q0v.hi -= v.hi + (u_q0v.lo < v.lo); u_q0v.lo -= v.lo; } *r_hi = u_q0v.hi; *r_lo = u_q0v.lo; LIBDIVIDE_ASSERT(q0.hi == 0); return q0.lo; #endif } ////////// UINT32 static inline struct libdivide_u32_t libdivide_internal_u32_gen(uint32_t d, int branchfree) { if (d == 0) { LIBDIVIDE_ERROR("divider must be != 0"); } struct libdivide_u32_t result; uint32_t floor_log_2_d = 31 - libdivide__count_leading_zeros32(d); if ((d & (d - 1)) == 0) { // Power of 2 if (! branchfree) { result.magic = 0; result.more = floor_log_2_d | LIBDIVIDE_U32_SHIFT_PATH; } else { // We want a magic number of 2**32 and a shift of floor_log_2_d // but one of the shifts is taken up by LIBDIVIDE_ADD_MARKER, // so we subtract 1 from the shift result.magic = 0; result.more = (floor_log_2_d-1) | LIBDIVIDE_ADD_MARKER; } } else { uint8_t more; uint32_t rem, proposed_m; proposed_m = libdivide_64_div_32_to_32(1U << floor_log_2_d, 0, d, &rem); LIBDIVIDE_ASSERT(rem > 0 && rem < d); const uint32_t e = d - rem; // This power works if e < 2**floor_log_2_d. if (!branchfree && (e < (1U << floor_log_2_d))) { // This power works more = floor_log_2_d; } else { // We have to use the general 33-bit algorithm. We need to compute // (2**power) / d. However, we already have (2**(power-1))/d and // its remainder. By doubling both, and then correcting the // remainder, we can compute the larger division. // don't care about overflow here - in fact, we expect it proposed_m += proposed_m; const uint32_t twice_rem = rem + rem; if (twice_rem >= d || twice_rem < rem) proposed_m += 1; more = floor_log_2_d | LIBDIVIDE_ADD_MARKER; } result.magic = 1 + proposed_m; result.more = more; // result.more's shift should in general be ceil_log_2_d. But if we // used the smaller power, we subtract one from the shift because we're // using the smaller power. If we're using the larger power, we // subtract one from the shift because it's taken care of by the add // indicator. So floor_log_2_d happens to be correct in both cases. } return result; } struct libdivide_u32_t libdivide_u32_gen(uint32_t d) { return libdivide_internal_u32_gen(d, 0); } struct libdivide_u32_branchfree_t libdivide_u32_branchfree_gen(uint32_t d) { if (d == 1) { LIBDIVIDE_ERROR("branchfree divider must be != 1"); } struct libdivide_u32_t tmp = libdivide_internal_u32_gen(d, 1); struct libdivide_u32_branchfree_t ret = {tmp.magic, (uint8_t)(tmp.more & LIBDIVIDE_32_SHIFT_MASK)}; return ret; } uint32_t libdivide_u32_do(uint32_t numer, const struct libdivide_u32_t *denom) { uint8_t more = denom->more; if (more & LIBDIVIDE_U32_SHIFT_PATH) { return numer >> (more & LIBDIVIDE_32_SHIFT_MASK); } else { uint32_t q = libdivide__mullhi_u32(denom->magic, numer); if (more & LIBDIVIDE_ADD_MARKER) { uint32_t t = ((numer - q) >> 1) + q; return t >> (more & LIBDIVIDE_32_SHIFT_MASK); } else { // all upper bits are 0 - don't need to mask them off return q >> more; } } } uint32_t libdivide_u32_recover(const struct libdivide_u32_t *denom) { uint8_t more = denom->more; uint8_t shift = more & LIBDIVIDE_32_SHIFT_MASK; if (more & LIBDIVIDE_U32_SHIFT_PATH) { return 1U << shift; } else if (!(more & LIBDIVIDE_ADD_MARKER)) { // We compute q = n/d = n*m / 2^(32 + shift) // Therefore we have d = 2^(32 + shift) / m // We need to ceil it. // We know d is not a power of 2, so m is not a power of 2, // so we can just add 1 to the floor uint32_t hi_dividend = 1U << shift; uint32_t rem_ignored; return 1 + libdivide_64_div_32_to_32(hi_dividend, 0, denom->magic, &rem_ignored); } else { // Here we wish to compute d = 2^(32+shift+1)/(m+2^32). // Notice (m + 2^32) is a 33 bit number. Use 64 bit division for now // Also note that shift may be as high as 31, so shift + 1 will // overflow. So we have to compute it as 2^(32+shift)/(m+2^32), and // then double the quotient and remainder. uint64_t half_n = 1ULL << (32 + shift); uint64_t d = (1ULL << 32) | denom->magic; // Note that the quotient is guaranteed <= 32 bits, but the remainder // may need 33! uint32_t half_q = (uint32_t)(half_n / d); uint64_t rem = half_n % d; // We computed 2^(32+shift)/(m+2^32) // Need to double it, and then add 1 to the quotient if doubling th // remainder would increase the quotient. // Note that rem<<1 cannot overflow, since rem < d and d is 33 bits uint32_t full_q = half_q + half_q + ((rem<<1) >= d); // We rounded down in gen unless we're a power of 2 (i.e. in branchfree case) // We can detect that by looking at m. If m zero, we're a power of 2 return full_q + (denom->magic != 0); } } uint32_t libdivide_u32_branchfree_recover(const struct libdivide_u32_branchfree_t *denom) { struct libdivide_u32_t denom_u32 = {denom->magic, (uint8_t)(denom->more | LIBDIVIDE_ADD_MARKER)}; return libdivide_u32_recover(&denom_u32); } int libdivide_u32_get_algorithm(const struct libdivide_u32_t *denom) { uint8_t more = denom->more; if (more & LIBDIVIDE_U32_SHIFT_PATH) return 0; else if (!(more & LIBDIVIDE_ADD_MARKER)) return 1; else return 2; } uint32_t libdivide_u32_do_alg0(uint32_t numer, const struct libdivide_u32_t *denom) { return numer >> (denom->more & LIBDIVIDE_32_SHIFT_MASK); } uint32_t libdivide_u32_do_alg1(uint32_t numer, const struct libdivide_u32_t *denom) { uint32_t q = libdivide__mullhi_u32(denom->magic, numer); return q >> denom->more; } uint32_t libdivide_u32_do_alg2(uint32_t numer, const struct libdivide_u32_t *denom) { // denom->add != 0 uint32_t q = libdivide__mullhi_u32(denom->magic, numer); uint32_t t = ((numer - q) >> 1) + q; // Note that this mask is typically free. Only the low bits are meaningful // to a shift, so compilers can optimize out this AND. return t >> (denom->more & LIBDIVIDE_32_SHIFT_MASK); } // same as algo 2 uint32_t libdivide_u32_branchfree_do(uint32_t numer, const struct libdivide_u32_branchfree_t *denom) { uint32_t q = libdivide__mullhi_u32(denom->magic, numer); uint32_t t = ((numer - q) >> 1) + q; return t >> denom->more; } #if defined(LIBDIVIDE_USE_SSE2) __m128i libdivide_u32_do_vector(__m128i numers, const struct libdivide_u32_t *denom) { uint8_t more = denom->more; if (more & LIBDIVIDE_U32_SHIFT_PATH) { return _mm_srl_epi32(numers, libdivide_u32_to_m128i(more & LIBDIVIDE_32_SHIFT_MASK)); } else { __m128i q = libdivide__mullhi_u32_flat_vector(numers, _mm_set1_epi32(denom->magic)); if (more & LIBDIVIDE_ADD_MARKER) { // uint32_t t = ((numer - q) >> 1) + q; // return t >> denom->shift; __m128i t = _mm_add_epi32(_mm_srli_epi32(_mm_sub_epi32(numers, q), 1), q); return _mm_srl_epi32(t, libdivide_u32_to_m128i(more & LIBDIVIDE_32_SHIFT_MASK)); } else { // q >> denom->shift return _mm_srl_epi32(q, libdivide_u32_to_m128i(more)); } } } __m128i libdivide_u32_do_vector_alg0(__m128i numers, const struct libdivide_u32_t *denom) { return _mm_srl_epi32(numers, libdivide_u32_to_m128i(denom->more & LIBDIVIDE_32_SHIFT_MASK)); } __m128i libdivide_u32_do_vector_alg1(__m128i numers, const struct libdivide_u32_t *denom) { __m128i q = libdivide__mullhi_u32_flat_vector(numers, _mm_set1_epi32(denom->magic)); return _mm_srl_epi32(q, libdivide_u32_to_m128i(denom->more)); } __m128i libdivide_u32_do_vector_alg2(__m128i numers, const struct libdivide_u32_t *denom) { __m128i q = libdivide__mullhi_u32_flat_vector(numers, _mm_set1_epi32(denom->magic)); __m128i t = _mm_add_epi32(_mm_srli_epi32(_mm_sub_epi32(numers, q), 1), q); return _mm_srl_epi32(t, libdivide_u32_to_m128i(denom->more & LIBDIVIDE_32_SHIFT_MASK)); } // same as algo 2 LIBDIVIDE_API __m128i libdivide_u32_branchfree_do_vector(__m128i numers, const struct libdivide_u32_branchfree_t *denom) { __m128i q = libdivide__mullhi_u32_flat_vector(numers, _mm_set1_epi32(denom->magic)); __m128i t = _mm_add_epi32(_mm_srli_epi32(_mm_sub_epi32(numers, q), 1), q); return _mm_srl_epi32(t, libdivide_u32_to_m128i(denom->more)); } #endif /////////// UINT64 static inline struct libdivide_u64_t libdivide_internal_u64_gen(uint64_t d, int branchfree) { if (d == 0) { LIBDIVIDE_ERROR("divider must be != 0"); } struct libdivide_u64_t result; uint32_t floor_log_2_d = 63 - libdivide__count_leading_zeros64(d); if ((d & (d - 1)) == 0) { // Power of 2 if (! branchfree) { result.magic = 0; result.more = floor_log_2_d | LIBDIVIDE_U64_SHIFT_PATH; } else { // We want a magic number of 2**64 and a shift of floor_log_2_d // but one of the shifts is taken up by LIBDIVIDE_ADD_MARKER, // so we subtract 1 from the shift result.magic = 0; result.more = (floor_log_2_d-1) | LIBDIVIDE_ADD_MARKER; } } else { uint64_t proposed_m, rem; uint8_t more; // (1 << (64 + floor_log_2_d)) / d proposed_m = libdivide_128_div_64_to_64(1ULL << floor_log_2_d, 0, d, &rem); LIBDIVIDE_ASSERT(rem > 0 && rem < d); const uint64_t e = d - rem; // This power works if e < 2**floor_log_2_d. if (!branchfree && e < (1ULL << floor_log_2_d)) { // This power works more = floor_log_2_d; } else { // We have to use the general 65-bit algorithm. We need to compute // (2**power) / d. However, we already have (2**(power-1))/d and // its remainder. By doubling both, and then correcting the // remainder, we can compute the larger division. // don't care about overflow here - in fact, we expect it proposed_m += proposed_m; const uint64_t twice_rem = rem + rem; if (twice_rem >= d || twice_rem < rem) proposed_m += 1; more = floor_log_2_d | LIBDIVIDE_ADD_MARKER; } result.magic = 1 + proposed_m; result.more = more; // result.more's shift should in general be ceil_log_2_d. But if we // used the smaller power, we subtract one from the shift because we're // using the smaller power. If we're using the larger power, we // subtract one from the shift because it's taken care of by the add // indicator. So floor_log_2_d happens to be correct in both cases, // which is why we do it outside of the if statement. } return result; } struct libdivide_u64_t libdivide_u64_gen(uint64_t d) { return libdivide_internal_u64_gen(d, 0); } struct libdivide_u64_branchfree_t libdivide_u64_branchfree_gen(uint64_t d) { if (d == 1) { LIBDIVIDE_ERROR("branchfree divider must be != 1"); } struct libdivide_u64_t tmp = libdivide_internal_u64_gen(d, 1); struct libdivide_u64_branchfree_t ret = {tmp.magic, (uint8_t)(tmp.more & LIBDIVIDE_64_SHIFT_MASK)}; return ret; } uint64_t libdivide_u64_do(uint64_t numer, const struct libdivide_u64_t *denom) { uint8_t more = denom->more; if (more & LIBDIVIDE_U64_SHIFT_PATH) { return numer >> (more & LIBDIVIDE_64_SHIFT_MASK); } else { uint64_t q = libdivide__mullhi_u64(denom->magic, numer); if (more & LIBDIVIDE_ADD_MARKER) { uint64_t t = ((numer - q) >> 1) + q; return t >> (more & LIBDIVIDE_64_SHIFT_MASK); } else { // all upper bits are 0 - don't need to mask them off return q >> more; } } } uint64_t libdivide_u64_recover(const struct libdivide_u64_t *denom) { uint8_t more = denom->more; uint8_t shift = more & LIBDIVIDE_64_SHIFT_MASK; if (more & LIBDIVIDE_U64_SHIFT_PATH) { return 1ULL << shift; } else if (!(more & LIBDIVIDE_ADD_MARKER)) { // We compute q = n/d = n*m / 2^(64 + shift) // Therefore we have d = 2^(64 + shift) / m // We need to ceil it. // We know d is not a power of 2, so m is not a power of 2, // so we can just add 1 to the floor uint64_t hi_dividend = 1ULL << shift; uint64_t rem_ignored; return 1 + libdivide_128_div_64_to_64(hi_dividend, 0, denom->magic, &rem_ignored); } else { // Here we wish to compute d = 2^(64+shift+1)/(m+2^64). // Notice (m + 2^64) is a 65 bit number. This gets hairy. See // libdivide_u32_recover for more on what we do here. // TODO: do something better than 128 bit math // Hack: if d is not a power of 2, this is a 128/128->64 divide // If d is a power of 2, this may be a bigger divide // However we can optimize that easily if (denom->magic == 0) { // 2^(64 + shift + 1) / (2^64) == 2^(shift + 1) return 1ULL << (shift + 1); } // Full n is a (potentially) 129 bit value // half_n is a 128 bit value // Compute the hi half of half_n. Low half is 0. uint64_t half_n_hi = 1ULL << shift, half_n_lo = 0; // d is a 65 bit value. The high bit is always set to 1. const uint64_t d_hi = 1, d_lo = denom->magic; // Note that the quotient is guaranteed <= 64 bits, // but the remainder may need 65! uint64_t r_hi, r_lo; uint64_t half_q = libdivide_128_div_128_to_64(half_n_hi, half_n_lo, d_hi, d_lo, &r_hi, &r_lo); // We computed 2^(64+shift)/(m+2^64) // Double the remainder ('dr') and check if that is larger than d // Note that d is a 65 bit value, so r1 is small and so r1 + r1 cannot // overflow uint64_t dr_lo = r_lo + r_lo; uint64_t dr_hi = r_hi + r_hi + (dr_lo < r_lo); // last term is carry int dr_exceeds_d = (dr_hi > d_hi) || (dr_hi == d_hi && dr_lo >= d_lo); uint64_t full_q = half_q + half_q + (dr_exceeds_d ? 1 : 0); return full_q + 1; } } uint64_t libdivide_u64_branchfree_recover(const struct libdivide_u64_branchfree_t *denom) { struct libdivide_u64_t denom_u64 = {denom->magic, (uint8_t)(denom->more | LIBDIVIDE_ADD_MARKER)}; return libdivide_u64_recover(&denom_u64); } int libdivide_u64_get_algorithm(const struct libdivide_u64_t *denom) { uint8_t more = denom->more; if (more & LIBDIVIDE_U64_SHIFT_PATH) return 0; else if (!(more & LIBDIVIDE_ADD_MARKER)) return 1; else return 2; } uint64_t libdivide_u64_do_alg0(uint64_t numer, const struct libdivide_u64_t *denom) { return numer >> (denom->more & LIBDIVIDE_64_SHIFT_MASK); } uint64_t libdivide_u64_do_alg1(uint64_t numer, const struct libdivide_u64_t *denom) { uint64_t q = libdivide__mullhi_u64(denom->magic, numer); return q >> denom->more; } uint64_t libdivide_u64_do_alg2(uint64_t numer, const struct libdivide_u64_t *denom) { uint64_t q = libdivide__mullhi_u64(denom->magic, numer); uint64_t t = ((numer - q) >> 1) + q; return t >> (denom->more & LIBDIVIDE_64_SHIFT_MASK); } // same as alg 2 uint64_t libdivide_u64_branchfree_do(uint64_t numer, const struct libdivide_u64_branchfree_t *denom) { uint64_t q = libdivide__mullhi_u64(denom->magic, numer); uint64_t t = ((numer - q) >> 1) + q; return t >> denom->more; } #if defined(LIBDIVIDE_USE_SSE2) __m128i libdivide_u64_do_vector(__m128i numers, const struct libdivide_u64_t *denom) { uint8_t more = denom->more; if (more & LIBDIVIDE_U64_SHIFT_PATH) { return _mm_srl_epi64(numers, libdivide_u32_to_m128i(more & LIBDIVIDE_64_SHIFT_MASK)); } else { __m128i q = libdivide_mullhi_u64_flat_vector(numers, libdivide__u64_to_m128(denom->magic)); if (more & LIBDIVIDE_ADD_MARKER) { // uint32_t t = ((numer - q) >> 1) + q; // return t >> denom->shift; __m128i t = _mm_add_epi64(_mm_srli_epi64(_mm_sub_epi64(numers, q), 1), q); return _mm_srl_epi64(t, libdivide_u32_to_m128i(more & LIBDIVIDE_64_SHIFT_MASK)); } else { // q >> denom->shift return _mm_srl_epi64(q, libdivide_u32_to_m128i(more)); } } } __m128i libdivide_u64_do_vector_alg0(__m128i numers, const struct libdivide_u64_t *denom) { return _mm_srl_epi64(numers, libdivide_u32_to_m128i(denom->more & LIBDIVIDE_64_SHIFT_MASK)); } __m128i libdivide_u64_do_vector_alg1(__m128i numers, const struct libdivide_u64_t *denom) { __m128i q = libdivide_mullhi_u64_flat_vector(numers, libdivide__u64_to_m128(denom->magic)); return _mm_srl_epi64(q, libdivide_u32_to_m128i(denom->more)); } __m128i libdivide_u64_do_vector_alg2(__m128i numers, const struct libdivide_u64_t *denom) { __m128i q = libdivide_mullhi_u64_flat_vector(numers, libdivide__u64_to_m128(denom->magic)); __m128i t = _mm_add_epi64(_mm_srli_epi64(_mm_sub_epi64(numers, q), 1), q); return _mm_srl_epi64(t, libdivide_u32_to_m128i(denom->more & LIBDIVIDE_64_SHIFT_MASK)); } __m128i libdivide_u64_branchfree_do_vector(__m128i numers, const struct libdivide_u64_branchfree_t *denom) { __m128i q = libdivide_mullhi_u64_flat_vector(numers, libdivide__u64_to_m128(denom->magic)); __m128i t = _mm_add_epi64(_mm_srli_epi64(_mm_sub_epi64(numers, q), 1), q); return _mm_srl_epi64(t, libdivide_u32_to_m128i(denom->more)); } #endif /////////// SINT32 static inline int32_t libdivide__mullhi_s32(int32_t x, int32_t y) { int64_t xl = x, yl = y; int64_t rl = xl * yl; // needs to be arithmetic shift return (int32_t)(rl >> 32); } static inline struct libdivide_s32_t libdivide_internal_s32_gen(int32_t d, int branchfree) { if (d == 0) { LIBDIVIDE_ERROR("divider must be != 0"); } struct libdivide_s32_t result; // If d is a power of 2, or negative a power of 2, we have to use a shift. // This is especially important because the magic algorithm fails for -1. // To check if d is a power of 2 or its inverse, it suffices to check // whether its absolute value has exactly one bit set. This works even for // INT_MIN, because abs(INT_MIN) == INT_MIN, and INT_MIN has one bit set // and is a power of 2. uint32_t ud = (uint32_t)d; uint32_t absD = (d < 0) ? -ud : ud; uint32_t floor_log_2_d = 31 - libdivide__count_leading_zeros32(absD); // check if exactly one bit is set, // don't care if absD is 0 since that's divide by zero if ((absD & (absD - 1)) == 0) { // Branchfree and normal paths are exactly the same result.magic = 0; result.more = floor_log_2_d | (d < 0 ? LIBDIVIDE_NEGATIVE_DIVISOR : 0) | LIBDIVIDE_S32_SHIFT_PATH; } else { LIBDIVIDE_ASSERT(floor_log_2_d >= 1); uint8_t more; // the dividend here is 2**(floor_log_2_d + 31), so the low 32 bit word // is 0 and the high word is floor_log_2_d - 1 uint32_t rem, proposed_m; proposed_m = libdivide_64_div_32_to_32(1U << (floor_log_2_d - 1), 0, absD, &rem); const uint32_t e = absD - rem; // We are going to start with a power of floor_log_2_d - 1. // This works if works if e < 2**floor_log_2_d. if (!branchfree && e < (1U << floor_log_2_d)) { // This power works more = floor_log_2_d - 1; } else { // We need to go one higher. This should not make proposed_m // overflow, but it will make it negative when interpreted as an // int32_t. proposed_m += proposed_m; const uint32_t twice_rem = rem + rem; if (twice_rem >= absD || twice_rem < rem) proposed_m += 1; more = floor_log_2_d | LIBDIVIDE_ADD_MARKER; } proposed_m += 1; int32_t magic = (int32_t)proposed_m; // Mark if we are negative. Note we only negate the magic number in the // branchfull case. if (d < 0) { more |= LIBDIVIDE_NEGATIVE_DIVISOR; if (!branchfree) { magic = -magic; } } result.more = more; result.magic = magic; } return result; } LIBDIVIDE_API struct libdivide_s32_t libdivide_s32_gen(int32_t d) { return libdivide_internal_s32_gen(d, 0); } LIBDIVIDE_API struct libdivide_s32_branchfree_t libdivide_s32_branchfree_gen(int32_t d) { if (d == 1) { LIBDIVIDE_ERROR("branchfree divider must be != 1"); } if (d == -1) { LIBDIVIDE_ERROR("branchfree divider must be != -1"); } struct libdivide_s32_t tmp = libdivide_internal_s32_gen(d, 1); struct libdivide_s32_branchfree_t result = {tmp.magic, tmp.more}; return result; } int32_t libdivide_s32_do(int32_t numer, const struct libdivide_s32_t *denom) { uint8_t more = denom->more; if (more & LIBDIVIDE_S32_SHIFT_PATH) { uint32_t sign = (int8_t)more >> 7; uint8_t shifter = more & LIBDIVIDE_32_SHIFT_MASK; uint32_t uq = (uint32_t)(numer + ((numer >> 31) & ((1U << shifter) - 1))); int32_t q = (int32_t)uq; q = q >> shifter; q = (q ^ sign) - sign; return q; } else { uint32_t uq = (uint32_t)libdivide__mullhi_s32(denom->magic, numer); if (more & LIBDIVIDE_ADD_MARKER) { // must be arithmetic shift and then sign extend int32_t sign = (int8_t)more >> 7; // q += (more < 0 ? -numer : numer), casts to avoid UB uq += ((uint32_t)numer ^ sign) - sign; } int32_t q = (int32_t)uq; q >>= more & LIBDIVIDE_32_SHIFT_MASK; q += (q < 0); return q; } } int32_t libdivide_s32_branchfree_do(int32_t numer, const struct libdivide_s32_branchfree_t *denom) { uint8_t more = denom->more; uint8_t shift = more & LIBDIVIDE_32_SHIFT_MASK; // must be arithmetic shift and then sign extend int32_t sign = (int8_t)more >> 7; int32_t magic = denom->magic; int32_t q = libdivide__mullhi_s32(magic, numer); q += numer; // If q is non-negative, we have nothing to do // If q is negative, we want to add either (2**shift)-1 if d is a power of // 2, or (2**shift) if it is not a power of 2 uint32_t is_power_of_2 = !!(more & LIBDIVIDE_S32_SHIFT_PATH); uint32_t q_sign = (uint32_t)(q >> 31); q += q_sign & ((1 << shift) - is_power_of_2); // Now arithmetic right shift q >>= shift; // Negate if needed q = (q ^ sign) - sign; return q; } int32_t libdivide_s32_recover(const struct libdivide_s32_t *denom) { uint8_t more = denom->more; uint8_t shift = more & LIBDIVIDE_32_SHIFT_MASK; if (more & LIBDIVIDE_S32_SHIFT_PATH) { uint32_t absD = 1U << shift; if (more & LIBDIVIDE_NEGATIVE_DIVISOR) { absD = -absD; } return (int32_t)absD; } else { // Unsigned math is much easier // We negate the magic number only in the branchfull case, and we don't // know which case we're in. However we have enough information to // determine the correct sign of the magic number. The divisor was // negative if LIBDIVIDE_NEGATIVE_DIVISOR is set. If ADD_MARKER is set, // the magic number's sign is opposite that of the divisor. // We want to compute the positive magic number. int negative_divisor = (more & LIBDIVIDE_NEGATIVE_DIVISOR); int magic_was_negated = (more & LIBDIVIDE_ADD_MARKER) ? denom->magic > 0 : denom->magic < 0; // Handle the power of 2 case (including branchfree) if (denom->magic == 0) { int32_t result = 1 << shift; return negative_divisor ? -result : result; } uint32_t d = (uint32_t)(magic_was_negated ? -denom->magic : denom->magic); uint64_t n = 1ULL << (32 + shift); // this shift cannot exceed 30 uint32_t q = (uint32_t)(n / d); int32_t result = (int32_t)q; result += 1; return negative_divisor ? -result : result; } } int32_t libdivide_s32_branchfree_recover(const struct libdivide_s32_branchfree_t *denom) { return libdivide_s32_recover((const struct libdivide_s32_t *)denom); } int libdivide_s32_get_algorithm(const struct libdivide_s32_t *denom) { uint8_t more = denom->more; int positiveDivisor = !(more & LIBDIVIDE_NEGATIVE_DIVISOR); if (more & LIBDIVIDE_S32_SHIFT_PATH) return (positiveDivisor ? 0 : 1); else if (more & LIBDIVIDE_ADD_MARKER) return (positiveDivisor ? 2 : 3); else return 4; } int32_t libdivide_s32_do_alg0(int32_t numer, const struct libdivide_s32_t *denom) { uint8_t shifter = denom->more & LIBDIVIDE_32_SHIFT_MASK; int32_t q = numer + ((numer >> 31) & ((1U << shifter) - 1)); return q >> shifter; } int32_t libdivide_s32_do_alg1(int32_t numer, const struct libdivide_s32_t *denom) { uint8_t shifter = denom->more & LIBDIVIDE_32_SHIFT_MASK; int32_t q = numer + ((numer >> 31) & ((1U << shifter) - 1)); return - (q >> shifter); } int32_t libdivide_s32_do_alg2(int32_t numer, const struct libdivide_s32_t *denom) { int32_t q = libdivide__mullhi_s32(denom->magic, numer); q += numer; q >>= denom->more & LIBDIVIDE_32_SHIFT_MASK; q += (q < 0); return q; } int32_t libdivide_s32_do_alg3(int32_t numer, const struct libdivide_s32_t *denom) { int32_t q = libdivide__mullhi_s32(denom->magic, numer); q -= numer; q >>= denom->more & LIBDIVIDE_32_SHIFT_MASK; q += (q < 0); return q; } int32_t libdivide_s32_do_alg4(int32_t numer, const struct libdivide_s32_t *denom) { int32_t q = libdivide__mullhi_s32(denom->magic, numer); q >>= denom->more & LIBDIVIDE_32_SHIFT_MASK; q += (q < 0); return q; } #if defined(LIBDIVIDE_USE_SSE2) __m128i libdivide_s32_do_vector(__m128i numers, const struct libdivide_s32_t *denom) { uint8_t more = denom->more; if (more & LIBDIVIDE_S32_SHIFT_PATH) { uint32_t shifter = more & LIBDIVIDE_32_SHIFT_MASK; __m128i roundToZeroTweak = _mm_set1_epi32((1U << shifter) - 1); // could use _mm_srli_epi32 with an all -1 register __m128i q = _mm_add_epi32(numers, _mm_and_si128(_mm_srai_epi32(numers, 31), roundToZeroTweak)); //q = numer + ((numer >> 31) & roundToZeroTweak); q = _mm_sra_epi32(q, libdivide_u32_to_m128i(shifter)); // q = q >> shifter __m128i shiftMask = _mm_set1_epi32((int32_t)((int8_t)more >> 7)); // set all bits of shift mask = to the sign bit of more q = _mm_sub_epi32(_mm_xor_si128(q, shiftMask), shiftMask); // q = (q ^ shiftMask) - shiftMask; return q; } else { __m128i q = libdivide_mullhi_s32_flat_vector(numers, _mm_set1_epi32(denom->magic)); if (more & LIBDIVIDE_ADD_MARKER) { __m128i sign = _mm_set1_epi32((int32_t)(int8_t)more >> 7); // must be arithmetic shift q = _mm_add_epi32(q, _mm_sub_epi32(_mm_xor_si128(numers, sign), sign)); // q += ((numer ^ sign) - sign); } q = _mm_sra_epi32(q, libdivide_u32_to_m128i(more & LIBDIVIDE_32_SHIFT_MASK)); // q >>= shift q = _mm_add_epi32(q, _mm_srli_epi32(q, 31)); // q += (q < 0) return q; } } __m128i libdivide_s32_do_vector_alg0(__m128i numers, const struct libdivide_s32_t *denom) { uint8_t shifter = denom->more & LIBDIVIDE_32_SHIFT_MASK; __m128i roundToZeroTweak = _mm_set1_epi32((1U << shifter) - 1); __m128i q = _mm_add_epi32(numers, _mm_and_si128(_mm_srai_epi32(numers, 31), roundToZeroTweak)); return _mm_sra_epi32(q, libdivide_u32_to_m128i(shifter)); } __m128i libdivide_s32_do_vector_alg1(__m128i numers, const struct libdivide_s32_t *denom) { uint8_t shifter = denom->more & LIBDIVIDE_32_SHIFT_MASK; __m128i roundToZeroTweak = _mm_set1_epi32((1U << shifter) - 1); __m128i q = _mm_add_epi32(numers, _mm_and_si128(_mm_srai_epi32(numers, 31), roundToZeroTweak)); return _mm_sub_epi32(_mm_setzero_si128(), _mm_sra_epi32(q, libdivide_u32_to_m128i(shifter))); } __m128i libdivide_s32_do_vector_alg2(__m128i numers, const struct libdivide_s32_t *denom) { __m128i q = libdivide_mullhi_s32_flat_vector(numers, _mm_set1_epi32(denom->magic)); q = _mm_add_epi32(q, numers); q = _mm_sra_epi32(q, libdivide_u32_to_m128i(denom->more & LIBDIVIDE_32_SHIFT_MASK)); q = _mm_add_epi32(q, _mm_srli_epi32(q, 31)); return q; } __m128i libdivide_s32_do_vector_alg3(__m128i numers, const struct libdivide_s32_t *denom) { __m128i q = libdivide_mullhi_s32_flat_vector(numers, _mm_set1_epi32(denom->magic)); q = _mm_sub_epi32(q, numers); q = _mm_sra_epi32(q, libdivide_u32_to_m128i(denom->more & LIBDIVIDE_32_SHIFT_MASK)); q = _mm_add_epi32(q, _mm_srli_epi32(q, 31)); return q; } __m128i libdivide_s32_do_vector_alg4(__m128i numers, const struct libdivide_s32_t *denom) { uint8_t more = denom->more; __m128i q = libdivide_mullhi_s32_flat_vector(numers, _mm_set1_epi32(denom->magic)); q = _mm_sra_epi32(q, libdivide_u32_to_m128i(more & LIBDIVIDE_32_SHIFT_MASK)); //q >>= shift q = _mm_add_epi32(q, _mm_srli_epi32(q, 31)); // q += (q < 0) return q; } __m128i libdivide_s32_branchfree_do_vector(__m128i numers, const struct libdivide_s32_branchfree_t *denom) { int32_t magic = denom->magic; uint8_t more = denom->more; uint8_t shift = more & LIBDIVIDE_32_SHIFT_MASK; // must be arithmetic shift __m128i sign = _mm_set1_epi32((int32_t)(int8_t)more >> 7); // libdivide__mullhi_s32(numers, magic); __m128i q = libdivide_mullhi_s32_flat_vector(numers, _mm_set1_epi32(magic)); q = _mm_add_epi32(q, numers); // q += numers // If q is non-negative, we have nothing to do // If q is negative, we want to add either (2**shift)-1 if d is a power of // 2, or (2**shift) if it is not a power of 2 uint32_t is_power_of_2 = (magic == 0); __m128i q_sign = _mm_srai_epi32(q, 31); // q_sign = q >> 31 __m128i mask = _mm_set1_epi32((1 << shift) - is_power_of_2); q = _mm_add_epi32(q, _mm_and_si128(q_sign, mask)); // q = q + (q_sign & mask) q = _mm_srai_epi32(q, shift); //q >>= shift q = _mm_sub_epi32(_mm_xor_si128(q, sign), sign); // q = (q ^ sign) - sign return q; } #endif ///////////// SINT64 static inline struct libdivide_s64_t libdivide_internal_s64_gen(int64_t d, int branchfree) { if (d == 0) { LIBDIVIDE_ERROR("divider must be != 0"); } struct libdivide_s64_t result; // If d is a power of 2, or negative a power of 2, we have to use a shift. // This is especially important because the magic algorithm fails for -1. // To check if d is a power of 2 or its inverse, it suffices to check // whether its absolute value has exactly one bit set. This works even for // INT_MIN, because abs(INT_MIN) == INT_MIN, and INT_MIN has one bit set // and is a power of 2. uint64_t ud = (uint64_t)d; uint64_t absD = (d < 0) ? -ud : ud; uint32_t floor_log_2_d = 63 - libdivide__count_leading_zeros64(absD); // check if exactly one bit is set, // don't care if absD is 0 since that's divide by zero if ((absD & (absD - 1)) == 0) { // Branchfree and non-branchfree cases are the same result.magic = 0; result.more = floor_log_2_d | (d < 0 ? LIBDIVIDE_NEGATIVE_DIVISOR : 0); } else { // the dividend here is 2**(floor_log_2_d + 63), so the low 64 bit word // is 0 and the high word is floor_log_2_d - 1 uint8_t more; uint64_t rem, proposed_m; proposed_m = libdivide_128_div_64_to_64(1ULL << (floor_log_2_d - 1), 0, absD, &rem); const uint64_t e = absD - rem; // We are going to start with a power of floor_log_2_d - 1. // This works if works if e < 2**floor_log_2_d. if (!branchfree && e < (1ULL << floor_log_2_d)) { // This power works more = floor_log_2_d - 1; } else { // We need to go one higher. This should not make proposed_m // overflow, but it will make it negative when interpreted as an // int32_t. proposed_m += proposed_m; const uint64_t twice_rem = rem + rem; if (twice_rem >= absD || twice_rem < rem) proposed_m += 1; // note that we only set the LIBDIVIDE_NEGATIVE_DIVISOR bit if we // also set ADD_MARKER this is an annoying optimization that // enables algorithm #4 to avoid the mask. However we always set it // in the branchfree case more = floor_log_2_d | LIBDIVIDE_ADD_MARKER; } proposed_m += 1; int64_t magic = (int64_t)proposed_m; // Mark if we are negative if (d < 0) { more |= LIBDIVIDE_NEGATIVE_DIVISOR; if (!branchfree) { magic = -magic; } } result.more = more; result.magic = magic; } return result; } struct libdivide_s64_t libdivide_s64_gen(int64_t d) { return libdivide_internal_s64_gen(d, 0); } struct libdivide_s64_branchfree_t libdivide_s64_branchfree_gen(int64_t d) { if (d == 1) { LIBDIVIDE_ERROR("branchfree divider must be != 1"); } if (d == -1) { LIBDIVIDE_ERROR("branchfree divider must be != -1"); } struct libdivide_s64_t tmp = libdivide_internal_s64_gen(d, 1); struct libdivide_s64_branchfree_t ret = {tmp.magic, tmp.more}; return ret; } int64_t libdivide_s64_do(int64_t numer, const struct libdivide_s64_t *denom) { uint8_t more = denom->more; int64_t magic = denom->magic; if (magic == 0) { //shift path uint32_t shifter = more & LIBDIVIDE_64_SHIFT_MASK; uint64_t uq = (uint64_t)numer + ((numer >> 63) & ((1ULL << shifter) - 1)); int64_t q = (int64_t)uq; q = q >> shifter; // must be arithmetic shift and then sign-extend int64_t shiftMask = (int8_t)more >> 7; q = (q ^ shiftMask) - shiftMask; return q; } else { uint64_t uq = (uint64_t)libdivide__mullhi_s64(magic, numer); if (more & LIBDIVIDE_ADD_MARKER) { // must be arithmetic shift and then sign extend int64_t sign = (int8_t)more >> 7; uq += ((uint64_t)numer ^ sign) - sign; } int64_t q = (int64_t)uq; q >>= more & LIBDIVIDE_64_SHIFT_MASK; q += (q < 0); return q; } } int64_t libdivide_s64_branchfree_do(int64_t numer, const struct libdivide_s64_branchfree_t *denom) { uint8_t more = denom->more; uint32_t shift = more & LIBDIVIDE_64_SHIFT_MASK; // must be arithmetic shift and then sign extend int64_t sign = (int8_t)more >> 7; int64_t magic = denom->magic; int64_t q = libdivide__mullhi_s64(magic, numer); q += numer; // If q is non-negative, we have nothing to do. // If q is negative, we want to add either (2**shift)-1 if d is a power of // 2, or (2**shift) if it is not a power of 2. uint32_t is_power_of_2 = (magic == 0); uint64_t q_sign = (uint64_t)(q >> 63); q += q_sign & ((1ULL << shift) - is_power_of_2); // Arithmetic right shift q >>= shift; // Negate if needed q = (q ^ sign) - sign; return q; } int64_t libdivide_s64_recover(const struct libdivide_s64_t *denom) { uint8_t more = denom->more; uint8_t shift = more & LIBDIVIDE_64_SHIFT_MASK; if (denom->magic == 0) { // shift path uint64_t absD = 1ULL << shift; if (more & LIBDIVIDE_NEGATIVE_DIVISOR) { absD = -absD; } return (int64_t)absD; } else { // Unsigned math is much easier int negative_divisor = (more & LIBDIVIDE_NEGATIVE_DIVISOR); int magic_was_negated = (more & LIBDIVIDE_ADD_MARKER) ? denom->magic > 0 : denom->magic < 0; uint64_t d = (uint64_t)(magic_was_negated ? -denom->magic : denom->magic); uint64_t n_hi = 1ULL << shift, n_lo = 0; uint64_t rem_ignored; uint64_t q = libdivide_128_div_64_to_64(n_hi, n_lo, d, &rem_ignored); int64_t result = (int64_t)(q + 1); if (negative_divisor) { result = -result; } return result; } } int64_t libdivide_s64_branchfree_recover(const struct libdivide_s64_branchfree_t *denom) { return libdivide_s64_recover((const struct libdivide_s64_t *)denom); } int libdivide_s64_get_algorithm(const struct libdivide_s64_t *denom) { uint8_t more = denom->more; int positiveDivisor = !(more & LIBDIVIDE_NEGATIVE_DIVISOR); if (denom->magic == 0) return (positiveDivisor ? 0 : 1); // shift path else if (more & LIBDIVIDE_ADD_MARKER) return (positiveDivisor ? 2 : 3); else return 4; } int64_t libdivide_s64_do_alg0(int64_t numer, const struct libdivide_s64_t *denom) { uint32_t shifter = denom->more & LIBDIVIDE_64_SHIFT_MASK; int64_t q = numer + ((numer >> 63) & ((1ULL << shifter) - 1)); return q >> shifter; } int64_t libdivide_s64_do_alg1(int64_t numer, const struct libdivide_s64_t *denom) { // denom->shifter != -1 && demo->shiftMask != 0 uint32_t shifter = denom->more & LIBDIVIDE_64_SHIFT_MASK; int64_t q = numer + ((numer >> 63) & ((1ULL << shifter) - 1)); return - (q >> shifter); } int64_t libdivide_s64_do_alg2(int64_t numer, const struct libdivide_s64_t *denom) { int64_t q = libdivide__mullhi_s64(denom->magic, numer); q += numer; q >>= denom->more & LIBDIVIDE_64_SHIFT_MASK; q += (q < 0); return q; } int64_t libdivide_s64_do_alg3(int64_t numer, const struct libdivide_s64_t *denom) { int64_t q = libdivide__mullhi_s64(denom->magic, numer); q -= numer; q >>= denom->more & LIBDIVIDE_64_SHIFT_MASK; q += (q < 0); return q; } int64_t libdivide_s64_do_alg4(int64_t numer, const struct libdivide_s64_t *denom) { int64_t q = libdivide__mullhi_s64(denom->magic, numer); q >>= denom->more & LIBDIVIDE_64_SHIFT_MASK; q += (q < 0); return q; } #if defined(LIBDIVIDE_USE_SSE2) __m128i libdivide_s64_do_vector(__m128i numers, const struct libdivide_s64_t *denom) { uint8_t more = denom->more; int64_t magic = denom->magic; if (magic == 0) { // shift path uint32_t shifter = more & LIBDIVIDE_64_SHIFT_MASK; __m128i roundToZeroTweak = libdivide__u64_to_m128((1ULL << shifter) - 1); __m128i q = _mm_add_epi64(numers, _mm_and_si128(libdivide_s64_signbits(numers), roundToZeroTweak)); // q = numer + ((numer >> 63) & roundToZeroTweak); q = libdivide_s64_shift_right_vector(q, shifter); // q = q >> shifter __m128i shiftMask = _mm_set1_epi32((int32_t)((int8_t)more >> 7)); q = _mm_sub_epi64(_mm_xor_si128(q, shiftMask), shiftMask); // q = (q ^ shiftMask) - shiftMask; return q; } else { __m128i q = libdivide_mullhi_s64_flat_vector(numers, libdivide__u64_to_m128(magic)); if (more & LIBDIVIDE_ADD_MARKER) { __m128i sign = _mm_set1_epi32((int32_t)((int8_t)more >> 7)); // must be arithmetic shift q = _mm_add_epi64(q, _mm_sub_epi64(_mm_xor_si128(numers, sign), sign)); // q += ((numer ^ sign) - sign); } // q >>= denom->mult_path.shift q = libdivide_s64_shift_right_vector(q, more & LIBDIVIDE_64_SHIFT_MASK); q = _mm_add_epi64(q, _mm_srli_epi64(q, 63)); // q += (q < 0) return q; } } __m128i libdivide_s64_do_vector_alg0(__m128i numers, const struct libdivide_s64_t *denom) { uint32_t shifter = denom->more & LIBDIVIDE_64_SHIFT_MASK; __m128i roundToZeroTweak = libdivide__u64_to_m128((1ULL << shifter) - 1); __m128i q = _mm_add_epi64(numers, _mm_and_si128(libdivide_s64_signbits(numers), roundToZeroTweak)); q = libdivide_s64_shift_right_vector(q, shifter); return q; } __m128i libdivide_s64_do_vector_alg1(__m128i numers, const struct libdivide_s64_t *denom) { uint32_t shifter = denom->more & LIBDIVIDE_64_SHIFT_MASK; __m128i roundToZeroTweak = libdivide__u64_to_m128((1ULL << shifter) - 1); __m128i q = _mm_add_epi64(numers, _mm_and_si128(libdivide_s64_signbits(numers), roundToZeroTweak)); q = libdivide_s64_shift_right_vector(q, shifter); return _mm_sub_epi64(_mm_setzero_si128(), q); } __m128i libdivide_s64_do_vector_alg2(__m128i numers, const struct libdivide_s64_t *denom) { __m128i q = libdivide_mullhi_s64_flat_vector(numers, libdivide__u64_to_m128(denom->magic)); q = _mm_add_epi64(q, numers); q = libdivide_s64_shift_right_vector(q, denom->more & LIBDIVIDE_64_SHIFT_MASK); q = _mm_add_epi64(q, _mm_srli_epi64(q, 63)); // q += (q < 0) return q; } __m128i libdivide_s64_do_vector_alg3(__m128i numers, const struct libdivide_s64_t *denom) { __m128i q = libdivide_mullhi_s64_flat_vector(numers, libdivide__u64_to_m128(denom->magic)); q = _mm_sub_epi64(q, numers); q = libdivide_s64_shift_right_vector(q, denom->more & LIBDIVIDE_64_SHIFT_MASK); q = _mm_add_epi64(q, _mm_srli_epi64(q, 63)); // q += (q < 0) return q; } __m128i libdivide_s64_do_vector_alg4(__m128i numers, const struct libdivide_s64_t *denom) { __m128i q = libdivide_mullhi_s64_flat_vector(numers, libdivide__u64_to_m128(denom->magic)); q = libdivide_s64_shift_right_vector(q, denom->more & LIBDIVIDE_64_SHIFT_MASK); q = _mm_add_epi64(q, _mm_srli_epi64(q, 63)); return q; } __m128i libdivide_s64_branchfree_do_vector(__m128i numers, const struct libdivide_s64_branchfree_t *denom) { int64_t magic = denom->magic; uint8_t more = denom->more; uint8_t shift = more & LIBDIVIDE_64_SHIFT_MASK; // must be arithmetic shift __m128i sign = _mm_set1_epi32((int32_t)(int8_t)more >> 7); // libdivide__mullhi_s64(numers, magic); __m128i q = libdivide_mullhi_s64_flat_vector(numers, libdivide__u64_to_m128(magic)); q = _mm_add_epi64(q, numers); // q += numers // If q is non-negative, we have nothing to do. // If q is negative, we want to add either (2**shift)-1 if d is a power of // 2, or (2**shift) if it is not a power of 2. uint32_t is_power_of_2 = (magic == 0); __m128i q_sign = libdivide_s64_signbits(q); // q_sign = q >> 63 __m128i mask = libdivide__u64_to_m128((1ULL << shift) - is_power_of_2); q = _mm_add_epi64(q, _mm_and_si128(q_sign, mask)); // q = q + (q_sign & mask) q = libdivide_s64_shift_right_vector(q, shift); // q >>= shift q = _mm_sub_epi64(_mm_xor_si128(q, sign), sign); // q = (q ^ sign) - sign return q; } #endif /////////// C++ stuff #ifdef __cplusplus // Our divider struct is templated on both a type (like uint64_t) and an // algorithm index. BRANCHFULL is the default algorithm, BRANCHFREE is the // branchfree variant, and the indexed variants are for unswitching. enum { BRANCHFULL = -1, BRANCHFREE = -2, ALGORITHM0 = 0, ALGORITHM1 = 1, ALGORITHM2 = 2, ALGORITHM3 = 3, ALGORITHM4 = 4 }; namespace libdivide_internal { #if defined(LIBDIVIDE_USE_SSE2) #define MAYBE_VECTOR(X) X #define MAYBE_VECTOR_PARAM(X) __m128i vector_func(__m128i, const X *) #else #define MAYBE_VECTOR(X) 0 #define MAYBE_VECTOR_PARAM(X) int unused #endif // The following convenience macros are used to build a type of the base // divider class and give it as template arguments the C functions // related to the macro name and the macro type paramaters. #define BRANCHFULL_DIVIDER(INT, TYPE) \ typedef base<INT, \ libdivide_##TYPE##_t, \ libdivide_##TYPE##_gen, \ libdivide_##TYPE##_do, \ MAYBE_VECTOR(libdivide_##TYPE##_do_vector)> #define BRANCHFREE_DIVIDER(INT, TYPE) \ typedef base<INT, \ libdivide_##TYPE##_branchfree_t, \ libdivide_##TYPE##_branchfree_gen, \ libdivide_##TYPE##_branchfree_do, \ MAYBE_VECTOR(libdivide_##TYPE##_branchfree_do_vector)> #define ALGORITHM_DIVIDER(INT, TYPE, ALGO) \ typedef base<INT, \ libdivide_##TYPE##_t, \ libdivide_##TYPE##_gen, \ libdivide_##TYPE##_do_##ALGO, \ MAYBE_VECTOR(libdivide_##TYPE##_do_vector_##ALGO)> #define CRASH_DIVIDER(INT, TYPE) \ typedef base<INT, \ libdivide_##TYPE##_t, \ libdivide_##TYPE##_gen, \ libdivide_##TYPE##_crash, \ MAYBE_VECTOR(libdivide_##TYPE##_crash_vector)> // Base divider, provides storage for the actual divider. // @IntType: e.g. uint32_t // @DenomType: e.g. libdivide_u32_t // @gen_func(): e.g. libdivide_u32_gen // @do_func(): e.g. libdivide_u32_do // @MAYBE_VECTOR_PARAM: e.g. libdivide_u32_do_vector template<typename IntType, typename DenomType, DenomType gen_func(IntType), IntType do_func(IntType, const DenomType *), MAYBE_VECTOR_PARAM(DenomType)> struct base { // Storage for the actual divider DenomType denom; // Constructor that takes a divisor value, and applies the gen function base(IntType d) : denom(gen_func(d)) { } // Default constructor to allow uninitialized uses in e.g. arrays base() {} // Needed for unswitch base(const DenomType& d) : denom(d) { } IntType perform_divide(IntType val) const { return do_func(val, &denom); } #if defined(LIBDIVIDE_USE_SSE2) __m128i perform_divide_vector(__m128i val) const { return vector_func(val, &denom); } #endif }; // Functions that will never be called but are required to be able // to use unswitch in C++ template code. Unsigned has fewer algorithms // than signed i.e. alg3 and alg4 are not defined for unsigned. In // order to make templates compile we need to define unsigned alg3 and // alg4 as crash functions. uint32_t libdivide_u32_crash(uint32_t, const libdivide_u32_t *) { exit(-1); } uint64_t libdivide_u64_crash(uint64_t, const libdivide_u64_t *) { exit(-1); } #if defined(LIBDIVIDE_USE_SSE2) __m128i libdivide_u32_crash_vector(__m128i, const libdivide_u32_t *) { exit(-1); } __m128i libdivide_u64_crash_vector(__m128i, const libdivide_u64_t *) { exit(-1); } #endif template<typename T, int ALGO> struct dispatcher { }; // Templated dispatch using partial specialization template<> struct dispatcher<int32_t, BRANCHFULL> { BRANCHFULL_DIVIDER(int32_t, s32) divider; }; template<> struct dispatcher<int32_t, BRANCHFREE> { BRANCHFREE_DIVIDER(int32_t, s32) divider; }; template<> struct dispatcher<int32_t, ALGORITHM0> { ALGORITHM_DIVIDER(int32_t, s32, alg0) divider; }; template<> struct dispatcher<int32_t, ALGORITHM1> { ALGORITHM_DIVIDER(int32_t, s32, alg1) divider; }; template<> struct dispatcher<int32_t, ALGORITHM2> { ALGORITHM_DIVIDER(int32_t, s32, alg2) divider; }; template<> struct dispatcher<int32_t, ALGORITHM3> { ALGORITHM_DIVIDER(int32_t, s32, alg3) divider; }; template<> struct dispatcher<int32_t, ALGORITHM4> { ALGORITHM_DIVIDER(int32_t, s32, alg4) divider; }; template<> struct dispatcher<uint32_t, BRANCHFULL> { BRANCHFULL_DIVIDER(uint32_t, u32) divider; }; template<> struct dispatcher<uint32_t, BRANCHFREE> { BRANCHFREE_DIVIDER(uint32_t, u32) divider; }; template<> struct dispatcher<uint32_t, ALGORITHM0> { ALGORITHM_DIVIDER(uint32_t, u32, alg0) divider; }; template<> struct dispatcher<uint32_t, ALGORITHM1> { ALGORITHM_DIVIDER(uint32_t, u32, alg1) divider; }; template<> struct dispatcher<uint32_t, ALGORITHM2> { ALGORITHM_DIVIDER(uint32_t, u32, alg2) divider; }; template<> struct dispatcher<uint32_t, ALGORITHM3> { CRASH_DIVIDER(uint32_t, u32) divider; }; template<> struct dispatcher<uint32_t, ALGORITHM4> { CRASH_DIVIDER(uint32_t, u32) divider; }; template<> struct dispatcher<int64_t, BRANCHFULL> { BRANCHFULL_DIVIDER(int64_t, s64) divider; }; template<> struct dispatcher<int64_t, BRANCHFREE> { BRANCHFREE_DIVIDER(int64_t, s64) divider; }; template<> struct dispatcher<int64_t, ALGORITHM0> { ALGORITHM_DIVIDER (int64_t, s64, alg0) divider; }; template<> struct dispatcher<int64_t, ALGORITHM1> { ALGORITHM_DIVIDER (int64_t, s64, alg1) divider; }; template<> struct dispatcher<int64_t, ALGORITHM2> { ALGORITHM_DIVIDER (int64_t, s64, alg2) divider; }; template<> struct dispatcher<int64_t, ALGORITHM3> { ALGORITHM_DIVIDER (int64_t, s64, alg3) divider; }; template<> struct dispatcher<int64_t, ALGORITHM4> { ALGORITHM_DIVIDER (int64_t, s64, alg4) divider; }; template<> struct dispatcher<uint64_t, BRANCHFULL> { BRANCHFULL_DIVIDER(uint64_t, u64) divider; }; template<> struct dispatcher<uint64_t, BRANCHFREE> { BRANCHFREE_DIVIDER(uint64_t, u64) divider; }; template<> struct dispatcher<uint64_t, ALGORITHM0> { ALGORITHM_DIVIDER(uint64_t, u64, alg0) divider; }; template<> struct dispatcher<uint64_t, ALGORITHM1> { ALGORITHM_DIVIDER(uint64_t, u64, alg1) divider; }; template<> struct dispatcher<uint64_t, ALGORITHM2> { ALGORITHM_DIVIDER(uint64_t, u64, alg2) divider; }; template<> struct dispatcher<uint64_t, ALGORITHM3> { CRASH_DIVIDER(uint64_t, u64) divider; }; template<> struct dispatcher<uint64_t, ALGORITHM4> { CRASH_DIVIDER(uint64_t, u64) divider; }; // Overloads that don't depend on the algorithm inline int32_t recover(const libdivide_s32_t *s) { return libdivide_s32_recover(s); } inline uint32_t recover(const libdivide_u32_t *s) { return libdivide_u32_recover(s); } inline int64_t recover(const libdivide_s64_t *s) { return libdivide_s64_recover(s); } inline uint64_t recover(const libdivide_u64_t *s) { return libdivide_u64_recover(s); } inline int32_t recover(const libdivide_s32_branchfree_t *s) { return libdivide_s32_branchfree_recover(s); } inline uint32_t recover(const libdivide_u32_branchfree_t *s) { return libdivide_u32_branchfree_recover(s); } inline int64_t recover(const libdivide_s64_branchfree_t *s) { return libdivide_s64_branchfree_recover(s); } inline uint64_t recover(const libdivide_u64_branchfree_t *s) { return libdivide_u64_branchfree_recover(s); } inline int get_algorithm(const libdivide_s32_t *s) { return libdivide_s32_get_algorithm(s); } inline int get_algorithm(const libdivide_u32_t *s) { return libdivide_u32_get_algorithm(s); } inline int get_algorithm(const libdivide_s64_t *s) { return libdivide_s64_get_algorithm(s); } inline int get_algorithm(const libdivide_u64_t *s) { return libdivide_u64_get_algorithm(s); } // Fallback for branchfree variants, which do not support unswitching template<typename T> int get_algorithm(const T *) { return -1; } } // This is the main divider class for use by the user (C++ API). // The divider itself is stored in the div variable who's // type is chosen by the dispatcher based on the template paramaters. template<typename T, int ALGO = BRANCHFULL> class divider { private: // Here's the actual divider typedef typename libdivide_internal::dispatcher<T, ALGO>::divider div_t; div_t div; // unswitch() friend declaration template<int NEW_ALGO, typename S> friend divider<S, NEW_ALGO> unswitch(const divider<S, BRANCHFULL> & d); // Constructor used by the unswitch friend divider(const div_t& denom) : div(denom) { } public: // Ordinary constructor that takes the divisor as a parameter divider(T n) : div(n) { } // Default constructor. We leave this deliberately undefined so that // creating an array of divider and then initializing them // doesn't slow us down. divider() { } // Divides the parameter by the divisor, returning the quotient T perform_divide(T val) const { return div.perform_divide(val); } // Recovers the divisor that was used to initialize the divider T recover_divisor() const { return libdivide_internal::recover(&div.denom); } #if defined(LIBDIVIDE_USE_SSE2) // Treats the vector as either two or four packed values (depending on the // size), and divides each of them by the divisor, // returning the packed quotients. __m128i perform_divide_vector(__m128i val) const { return div.perform_divide_vector(val); } #endif // Returns the index of algorithm, for use in the unswitch function. Does // not apply to branchfree variant. // Returns the algorithm for unswitching. int get_algorithm() const { return libdivide_internal::get_algorithm(&div.denom); } bool operator==(const divider<T, ALGO>& him) const { return div.denom.magic == him.div.denom.magic && div.denom.more == him.div.denom.more; } bool operator!=(const divider<T, ALGO>& him) const { return !(*this == him); } }; #if __cplusplus >= 201103L || \ (defined(_MSC_VER) && _MSC_VER >= 1800) // libdivdie::branchfree_divider<T> template <typename T> using branchfree_divider = divider<T, BRANCHFREE>; #endif // Returns a divider specialized for the given algorithm template<int NEW_ALGO, typename T> divider<T, NEW_ALGO> unswitch(const divider<T, BRANCHFULL>& d) { return divider<T, NEW_ALGO>(d.div.denom); } // Overload of the / operator for scalar division template<typename int_type, int ALGO> int_type operator/(int_type numer, const divider<int_type, ALGO>& denom) { return denom.perform_divide(numer); } // Overload of the /= operator for scalar division template<typename int_type, int ALGO> int_type operator/=(int_type& numer, const divider<int_type, ALGO>& denom) { numer = denom.perform_divide(numer); return numer; } #if defined(LIBDIVIDE_USE_SSE2) // Overload of the / operator for vector division template<typename int_type, int ALGO> __m128i operator/(__m128i numer, const divider<int_type, ALGO>& denom) { return denom.perform_divide_vector(numer); } // Overload of the /= operator for vector division template<typename int_type, int ALGO> __m128i operator/=(__m128i& numer, const divider<int_type, ALGO>& denom) { numer = denom.perform_divide_vector(numer); return numer; } #endif } // namespace libdivide } // anonymous namespace #endif // __cplusplus #endif // LIBDIVIDE_H #ifndef LOCAL #pragma GCC optimize ("O3") #endif #include <bits/stdc++.h> using namespace std; #define sim template < class c #define ris return * this #define dor > debug & operator << #define eni(x) sim > typename \ enable_if<sizeof dud<c>(0) x 1, debug&>::type operator<<(c i) { sim > struct rge { c b, e; }; sim > rge<c> range(c i, c j) { return {i, j}; } sim > auto dud(c* x) -> decltype(cerr << *x, 0); sim > char dud(...); struct debug { #ifdef LOCAL ~debug() { cerr << endl; } eni(!=) cerr << boolalpha << i; ris; } eni(==) ris << range(begin(i), end(i)); } sim, class b dor(pair < b, c > d) { ris << "(" << d.first << ", " << d.second << ")"; } sim dor(rge<c> d) { *this << "["; for (c it = d.b; it != d.e; ++it) *this << ", " + 2 * (it == d.b) << *it; ris << "]"; } #else sim dor(const c&) { ris; } #endif }; #define imie(x...) " [" #x ": " << (x) << "] " #include <ext/pb_ds/assoc_container.hpp> #include <ext/pb_ds/tree_policy.hpp> template <typename A, typename B> using unordered_map2 = __gnu_pbds::gp_hash_table<A, B>; using namespace __gnu_pbds; template <typename T> using ordered_set = __gnu_pbds::tree<T, __gnu_pbds::null_type, less<T>, __gnu_pbds::rb_tree_tag, __gnu_pbds::tree_order_statistics_node_update>; // ordered_set<int> s; s.insert(1); s.insert(2); // s.order_of_key(1); // Out: 0. // *s.find_by_order(1); // Out: 2. using ld = long double; using ll = long long; int mod = 1000 * 1000 * 1000 + 7; libdivide::libdivide_u64_t fast_mod; int Moduluj(uint64_t x) { return x - mod * libdivide::libdivide_u64_do(x, &fast_mod); } void OdejmijOd(int& a, int b) { a -= b; if (a < 0) a += mod; } int Odejmij(int a, int b) { OdejmijOd(a, b); return a; } void DodajDo(int& a, int b) { a += b; if (a >= mod) a -= mod; } int Dodaj(int a, int b) { DodajDo(a, b); return a; } int Mnoz(int a, int b) { return Moduluj(uint64_t(a) * b); } void MnozDo(int& a, int b) { a = Mnoz(a, b); } int Pot(int a, ll b) { int res = 1; while (b) { if (b % 2 == 1) MnozDo(res, a); a = Mnoz(a, a); b /= 2; } return res; } int Odw(int a) { return Pot(a, mod - 2); } void PodzielDo(int& a, int b) { MnozDo(a, Odw(b)); } int Podziel(int a, int b) { return Mnoz(a, Odw(b)); } template <typename T> T Maxi(T& a, T b) { return a = max(a, b); } template <typename T> T Mini(T& a, T b) { return a = min(a, b); } constexpr int nax = 1005; constexpr int kax = 12; vector<int> Pousuwaj(const vector<int>& v) { vector<int> res; const int n = (int) v.size(); for (int i = 0; i < n; i++) { int ile = 0; if (i > 0 and v[i - 1] > v[i]) ile++; if (i + 1 < n and v[i + 1] > v[i]) ile++; if (ile == 0) { res.push_back(v[i]); } } return res; } int Licz(int ile, const vector<int>& v) { if (ile == 0) { if (v.empty()) return 0; if ((int) v.size() == 1) return 1; return Licz(0, Pousuwaj(v)) + 1; } else if (ile == 1) { assert(!v.empty()); if ((int) v.size() == 1) return 0; return Licz(1, Pousuwaj(v)) + 1; } else if (ile == 2) { assert((int) v.size() >= 2); if ((int) v.size() == 2) return 0; return Licz(2, Pousuwaj(v)) + 1; } else { assert(false); } } int Brut(int n, int k, int lewo, int prawo) { const int ile = lewo + prawo; vector<int> v(n); iota(v.begin(), v.end(), 0); int a = 0, b = n; if (ile == 0) { // nic. } else if (ile == 1) { v.push_back(numeric_limits<int>::max()); } else if (ile == 2) { v.insert(v.begin(), numeric_limits<int>::max()); v.push_back(numeric_limits<int>::max() - 1); a++; b++; } else { assert(false); } int wyn = 0; map<int, int> licz; do { //if (n == 3 and k == 2) debug() << imie(v); if (Licz(ile, v) == k) { if (n == 6 and k == 3 and lewo and !prawo) { licz[max_element(v.begin() + a, v.begin() + b) - v.begin()]++; } wyn++; } } while (next_permutation(v.begin() + a, v.begin() + b)); if (n == 6 and k == 3 and lewo and !prawo) { debug() << imie(n) imie(k) imie(lewo) imie(prawo) imie(licz); } return Moduluj(wyn); } int C[nax][nax]; int Dp(int n, int k, int lewo, int prawo); int Pref(int n, int k, int lewo, int prawo); int Pref_(int n, int k, int lewo, int prawo) { return Dodaj(Dp(n, k, lewo, prawo), Pref(n, k - 1, lewo, prawo)); } int Pref(int n, int k, int lewo, int prawo) { if (n < 0 or k < 0) return 0; if (lewo > prawo) swap(lewo, prawo); static int dp[nax][kax][2][2]; static bool juz[nax][kax][2][2]; if (juz[n][k][lewo][prawo]) { return dp[n][k][lewo][prawo]; } juz[n][k][lewo][prawo] = true; return dp[n][k][lewo][prawo] = Pref_(n, k, lewo, prawo); } int Dp_(int n, int k, int lewo, int prawo) { if (n == 0) { return k == 0; } if (n == 1) { return k == 1; } int res = 0; if (lewo) { DodajDo(res, Dp(n - 1, k, 1, prawo)); } else { DodajDo(res, Dp(n - 1, k - 1, 1, prawo)); } if (prawo) { DodajDo(res, Dp(n - 1, k, lewo, 1)); } else { DodajDo(res, Dp(n - 1, k - 1, lewo, 1)); } for (int i = 2; i < n; i++) { const int a = i - 1; const int b = n - i; int ile = 0; if (lewo == 0 and prawo == 0) { DodajDo(ile, Mnoz(Dp(a, k - 1, lewo, 1), Pref(b, k - 1, 1, prawo))); DodajDo(ile, Mnoz(Pref(a, k - 1, lewo, 1), Dp(b, k - 1, 1, prawo))); OdejmijOd(ile, Mnoz(Dp(a, k - 1, lewo, 1), Dp(b, k - 1, 1, prawo))); } else if (lewo == 1 and prawo == 1) { DodajDo(ile, Mnoz(Dp(a, k - 1, lewo, 1), Dp(b, k - 1, 1, prawo))); DodajDo(ile, Mnoz(Dp(a, k, lewo, 1), Pref(b, k - 1, 1, prawo))); DodajDo(ile, Mnoz(Pref(a, k - 1, lewo, 1), Dp(b, k, 1, prawo))); } else if (lewo == 1) { DodajDo(ile, Mnoz(Dp(a, k - 1, lewo, 1), Pref(b, k, 1, prawo))); DodajDo(ile, Mnoz(Pref(a, k - 2, lewo, 1), Dp(b, k, 1, prawo))); } else if (prawo == 1) { DodajDo(ile, Mnoz(Pref(a, k, lewo, 1), Dp(b, k - 1, 1, prawo))); DodajDo(ile, Mnoz(Dp(a, k, lewo, 1), Pref(b, k - 2, 1, prawo))); } else { assert(false); } // DodajDo(ile, Mnoz(Dp(a, k - 1, lewo, 1), Pref(b, k - 1, 1, prawo))); // DodajDo(ile, Mnoz(Pref(a, k - 1, lewo, 1), Dp(b, k - 1, 1, prawo))); // OdejmijOd(ile, Mnoz(Dp(a, k - 1, lewo, 1), Dp(b, k - 1, 1, prawo))); // debug() << imie(a) imie(b) imie(lewo) imie(prawo); // debug() << imie(Dp(a, k - 1, lewo, 1)) imie(Pref(a, k - 1, lewo, 1)); // debug() << imie(Dp(b, k - 1, 1, prawo)) imie(Pref(b, k - 1, 1, prawo)); // debug() << imie(ile) imie(C[a + b][a]); DodajDo(res, Mnoz(ile, C[a + b][a])); } debug() << "Dp(" imie(n) imie(k) imie(lewo) imie(prawo) ") = " << res; #ifdef LOCAL // const int b = Brut(n, k, lewo, prawo); // if (b != res) { // debug() << imie(res) imie(b); // assert(false); // } #endif return res; } int Dp(int n, int k, int lewo, int prawo) { if (n < 0 or k < 0) return 0; if (lewo > prawo) swap(lewo, prawo); static int tab[nax][kax][2][2]; static bool juz[nax][kax][2][2]; if (juz[n][k][lewo][prawo]) { return tab[n][k][lewo][prawo]; } juz[n][k][lewo][prawo] = true; return tab[n][k][lewo][prawo] = Dp_(n, k, lewo, prawo); } int Daj(int n, int k) { if (!(1 <= n and n < nax and 1 <= k and k + 1 < kax)) return 0; const int res = Dp(n, k + 1, 0, 0); assert(res >= 0); return res; } int main() { ios_base::sync_with_stdio(0); cin.tie(0); int n, k; cin >> n >> k >> mod; fast_mod = libdivide::libdivide_u64_gen(mod); for (int A = 0; A < nax; A++) { C[A][0] = C[A][A] = 1; for (int B = 1; B < A; B++) { C[A][B] = Dodaj(C[A - 1][B - 1], C[A - 1][B]); } } cout << Daj(n, k) << endl; // while (cin >> n >> k) { // cout << Daj(n, k) << endl; // } // int suma = 0; // for (n = 1; n <= 1000; n++) { // for (k = 1; k <= n; k++) { // DodajDo(suma, Mnoz(Mnoz(n, k), Daj(n, k))); // } // } // cout << suma << endl; // // wynik: 59180009 // for (int n = 0; n <= 10; n++) { // for (int k = 0; k <= n; k++) { // Dp(n, k, 0, 0); // } // } // printf("[n = %2d]:", n_); // for (int i = 0; i <= n_; i++) { // printf(" %7d", Dp(n_, i, 0, 0)); // } // printf("\n"); return 0; } |