mathematics.c

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/*
 * Copyright (c) 2005-2012 Michael Niedermayer <michaelni@gmx.at>
 *
 * This file is part of FFmpeg.
 *
 * FFmpeg is free software; you can redistribute it and/or
 * modify it under the terms of the GNU Lesser General Public
 * License as published by the Free Software Foundation; either
 * version 2.1 of the License, or (at your option) any later version.
 *
 * FFmpeg is distributed in the hope that it will be useful,
 * but WITHOUT ANY WARRANTY; without even the implied warranty of
 * MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE.  See the GNU
 * Lesser General Public License for more details.
 *
 * You should have received a copy of the GNU Lesser General Public
 * License along with FFmpeg; if not, write to the Free Software
 * Foundation, Inc., 51 Franklin Street, Fifth Floor, Boston, MA 02110-1301 USA
 */
 
/**
 * @file
 * miscellaneous math routines and tables
 */
 
#include <stdint.h>
#include <limits.h>
 
#include "avutil.h"
#include "mathematics.h"
#include "libavutil/intmath.h"
#include "libavutil/common.h"
#include "avassert.h"
 
/* Stein's binary GCD algorithm:
 * https://en.wikipedia.org/wiki/Binary_GCD_algorithm */
int64_t av_gcd(int64_t a, int64_t b) {
    int za, zb, k;
    int64_t u, v;
    if (a == 0)
        return b;
    if (b == 0)
        return a;
    za = ff_ctzll(a);
    zb = ff_ctzll(b);
    k  = FFMIN(za, zb);
    u = llabs(a >> za);
    v = llabs(b >> zb);
    while (u != v) {
        if (u > v)
            FFSWAP(int64_t, v, u);
        v -= u;
        v >>= ff_ctzll(v);
    }
    return (uint64_t)u << k;
}
 
int64_t av_rescale_rnd(int64_t a, int64_t b, int64_t c, enum AVRounding rnd)
{
    int64_t r = 0;
    av_assert2(c > 0);
    av_assert2(b >=0);
    av_assert2((unsigned)(rnd&~AV_ROUND_PASS_MINMAX)<=5 && (rnd&~AV_ROUND_PASS_MINMAX)!=4);
 
    if (c <= 0 || b < 0 || !((unsigned)(rnd&~AV_ROUND_PASS_MINMAX)<=5 && (rnd&~AV_ROUND_PASS_MINMAX)!=4))
        return INT64_MIN;
 
    if (rnd & AV_ROUND_PASS_MINMAX) {
        if (a == INT64_MIN || a == INT64_MAX)
            return a;
        rnd -= AV_ROUND_PASS_MINMAX;
    }
 
    if (a < 0)
        return -(uint64_t)av_rescale_rnd(-FFMAX(a, -INT64_MAX), b, c, rnd ^ ((rnd >> 1) & 1));
 
    if (rnd == AV_ROUND_NEAR_INF)
        r = c / 2;
    else if (rnd & 1)
        r = c - 1;
 
    if (b <= INT_MAX && c <= INT_MAX) {
        if (a <= INT_MAX)
            return (a * b + r) / c;
        else {
            int64_t ad = a / c;
            int64_t a2 = (a % c * b + r) / c;
            if (ad >= INT32_MAX && b && ad > (INT64_MAX - a2) / b)
                return INT64_MIN;
            return ad * b + a2;
        }
    } else {
#if 1
        uint64_t a0  = a & 0xFFFFFFFF;
        uint64_t a1  = a >> 32;
        uint64_t b0  = b & 0xFFFFFFFF;
        uint64_t b1  = b >> 32;
        uint64_t t1  = a0 * b1 + a1 * b0;
        uint64_t t1a = t1 << 32;
        int i;
 
        a0  = a0 * b0 + t1a;
        a1  = a1 * b1 + (t1 >> 32) + (a0 < t1a);
        a0 += r;
        a1 += a0 < r;
 
        for (i = 63; i >= 0; i--) {
            a1 += a1 + ((a0 >> i) & 1);
            t1 += t1;
            if (c <= a1) {
                a1 -= c;
                t1++;
            }
        }
        if (t1 > INT64_MAX)
            return INT64_MIN;
        return t1;
#else
        /* reference code doing (a*b + r) / c, requires libavutil/integer.h */
        AVInteger ai;
        ai = av_mul_i(av_int2i(a), av_int2i(b));
        ai = av_add_i(ai, av_int2i(r));
 
        return av_i2int(av_div_i(ai, av_int2i(c)));
#endif
    }
}
 
int64_t av_rescale(int64_t a, int64_t b, int64_t c)
{
    return av_rescale_rnd(a, b, c, AV_ROUND_NEAR_INF);
}
 
int64_t av_rescale_q_rnd(int64_t a, AVRational bq, AVRational cq,
                         enum AVRounding rnd)
{
    int64_t b = bq.num * (int64_t)cq.den;
    int64_t c = cq.num * (int64_t)bq.den;
    return av_rescale_rnd(a, b, c, rnd);
}
 
int64_t av_rescale_q(int64_t a, AVRational bq, AVRational cq)
{
    return av_rescale_q_rnd(a, bq, cq, AV_ROUND_NEAR_INF);
}
 
int av_compare_ts(int64_t ts_a, AVRational tb_a, int64_t ts_b, AVRational tb_b)
{
    int64_t a = tb_a.num * (int64_t)tb_b.den;
    int64_t b = tb_b.num * (int64_t)tb_a.den;
    if ((FFABS64U(ts_a)|a|FFABS64U(ts_b)|b) <= INT_MAX)
        return (ts_a*a > ts_b*b) - (ts_a*a < ts_b*b);
    if (av_rescale_rnd(ts_a, a, b, AV_ROUND_DOWN) < ts_b)
        return -1;
    if (av_rescale_rnd(ts_b, b, a, AV_ROUND_DOWN) < ts_a)
        return 1;
    return 0;
}
 
int64_t av_compare_mod(uint64_t a, uint64_t b, uint64_t mod)
{
    int64_t c = (a - b) & (mod - 1);
    if (c > (mod >> 1))
        c -= mod;
    return c;
}
 
int64_t av_rescale_delta(AVRational in_tb, int64_t in_ts,  AVRational fs_tb, int duration, int64_t *last, AVRational out_tb){
    int64_t a, b, this;
 
    av_assert0(in_ts != AV_NOPTS_VALUE);
    av_assert0(duration >= 0);
 
    if (*last == AV_NOPTS_VALUE || !duration || in_tb.num*(int64_t)out_tb.den <= out_tb.num*(int64_t)in_tb.den) {
simple_round:
        *last = av_rescale_q(in_ts, in_tb, fs_tb) + duration;
        return av_rescale_q(in_ts, in_tb, out_tb);
    }
 
    a =  av_rescale_q_rnd(2*in_ts-1, in_tb, fs_tb, AV_ROUND_DOWN)   >>1;
    b = (av_rescale_q_rnd(2*in_ts+1, in_tb, fs_tb, AV_ROUND_UP  )+1)>>1;
    if (*last < 2*a - b || *last > 2*b - a)
        goto simple_round;
 
    this = av_clip64(*last, a, b);
    *last = this + duration;
 
    return av_rescale_q(this, fs_tb, out_tb);
}
 
int64_t av_add_stable(AVRational ts_tb, int64_t ts, AVRational inc_tb, int64_t inc)
{
    int64_t m, d;
 
    if (inc != 1)
        inc_tb = av_mul_q(inc_tb, (AVRational) {inc, 1});
 
    m = inc_tb.num * (int64_t)ts_tb.den;
    d = inc_tb.den * (int64_t)ts_tb.num;
 
    if (m % d == 0 && ts <= INT64_MAX - m / d)
        return ts + m / d;
    if (m < d)
        return ts;
 
    {
        int64_t old = av_rescale_q(ts, ts_tb, inc_tb);
        int64_t old_ts = av_rescale_q(old, inc_tb, ts_tb);
 
        if (old == INT64_MAX || old == AV_NOPTS_VALUE || old_ts == AV_NOPTS_VALUE)
            return ts;
 
        return av_sat_add64(av_rescale_q(old + 1, inc_tb, ts_tb), ts - old_ts);
    }
}
 
static inline double eval_poly(const double *coeff, int size, double x) {
    double sum = coeff[size-1];
    int i;
    for (i = size-2; i >= 0; --i) {
        sum *= x;
        sum += coeff[i];
    }
    return sum;
}
 
/**
 * 0th order modified bessel function of the first kind.
 * Algorithm taken from the Boost project, source:
 * https://searchcode.com/codesearch/view/14918379/
 * Use, modification and distribution are subject to the
 * Boost Software License, Version 1.0 (see notice below).
 * Boost Software License - Version 1.0 - August 17th, 2003
Permission is hereby granted, free of charge, to any person or organization
obtaining a copy of the software and accompanying documentation covered by
this license (the "Software") to use, reproduce, display, distribute,
execute, and transmit the Software, and to prepare derivative works of the
Software, and to permit third-parties to whom the Software is furnished to
do so, all subject to the following:
 
The copyright notices in the Software and this entire statement, including
the above license grant, this restriction and the following disclaimer,
must be included in all copies of the Software, in whole or in part, and
all derivative works of the Software, unless such copies or derivative
works are solely in the form of machine-executable object code generated by
a source language processor.
 
THE SOFTWARE IS PROVIDED "AS IS", WITHOUT WARRANTY OF ANY KIND, EXPRESS OR
IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES OF MERCHANTABILITY,
FITNESS FOR A PARTICULAR PURPOSE, TITLE AND NON-INFRINGEMENT. IN NO EVENT
SHALL THE COPYRIGHT HOLDERS OR ANYONE DISTRIBUTING THE SOFTWARE BE LIABLE
FOR ANY DAMAGES OR OTHER LIABILITY, WHETHER IN CONTRACT, TORT OR OTHERWISE,
ARISING FROM, OUT OF OR IN CONNECTION WITH THE SOFTWARE OR THE USE OR OTHER
DEALINGS IN THE SOFTWARE.
 */
 
double av_bessel_i0(double x) {
// Modified Bessel function of the first kind of order zero
// minimax rational approximations on intervals, see
// Blair and Edwards, Chalk River Report AECL-4928, 1974
    static const double p1[] = {
        -2.2335582639474375249e+15,
        -5.5050369673018427753e+14,
        -3.2940087627407749166e+13,
        -8.4925101247114157499e+11,
        -1.1912746104985237192e+10,
        -1.0313066708737980747e+08,
        -5.9545626019847898221e+05,
        -2.4125195876041896775e+03,
        -7.0935347449210549190e+00,
        -1.5453977791786851041e-02,
        -2.5172644670688975051e-05,
        -3.0517226450451067446e-08,
        -2.6843448573468483278e-11,
        -1.5982226675653184646e-14,
        -5.2487866627945699800e-18,
    };
    static const double q1[] = {
        -2.2335582639474375245e+15,
         7.8858692566751002988e+12,
        -1.2207067397808979846e+10,
         1.0377081058062166144e+07,
        -4.8527560179962773045e+03,
         1.0,
    };
    static const double p2[] = {
        -2.2210262233306573296e-04,
         1.3067392038106924055e-02,
        -4.4700805721174453923e-01,
         5.5674518371240761397e+00,
        -2.3517945679239481621e+01,
         3.1611322818701131207e+01,
        -9.6090021968656180000e+00,
    };
    static const double q2[] = {
        -5.5194330231005480228e-04,
         3.2547697594819615062e-02,
        -1.1151759188741312645e+00,
         1.3982595353892851542e+01,
        -6.0228002066743340583e+01,
         8.5539563258012929600e+01,
        -3.1446690275135491500e+01,
        1.0,
    };
    double y, r, factor;
    if (x == 0)
        return 1.0;
    x = fabs(x);
    if (x <= 15) {
        y = x * x;
        return eval_poly(p1, FF_ARRAY_ELEMS(p1), y) / eval_poly(q1, FF_ARRAY_ELEMS(q1), y);
    }
    else {
        y = 1 / x - 1.0 / 15;
        r = eval_poly(p2, FF_ARRAY_ELEMS(p2), y) / eval_poly(q2, FF_ARRAY_ELEMS(q2), y);
        factor = exp(x) / sqrt(x);
        return factor * r;
    }
}