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bcdmath.cpp
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bcdmath.cpp
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/**
*
* Copyright (c) 2010-2015 Voidware Ltd. All Rights Reserved.
*
* This file contains Original Code and/or Modifications of Original Code as
* defined in and that are subject to the Voidware Public Source Licence version
* 1.0 (the 'Licence'). You may not use this file except in compliance with the
* Licence or with expressly written permission from Voidware. Please obtain a
* copy of the Licence at http://www.voidware.com/legal/vpsl1.txt and read it
* before using this file.
*
* The Original Code and all software distributed under the Licence are
* distributed on an 'AS IS' basis, WITHOUT WARRANTY OF ANY KIND, EITHER EXPRESS
* OR IMPLIED, AND VOIDWARE HEREBY DISCLAIMS ALL SUCH WARRANTIES, INCLUDING
* WITHOUT LIMITATION, ANY WARRANTIES OF MERCHANTABILITY, FITNESS FOR A
* PARTICULAR PURPOSE, QUIET ENJOYMENT OR NON-INFRINGEMENT.
*
* Please see the Licence for the specific language governing rights and
* limitations under the Licence.
*
*/
#include <string.h> // memcpy
// test for nPI etc.
// this gives exact results for close to special values
// Casio do this but HP do not
#define TRIG_SPECIAL_CASESxx
#ifdef ARDUINO
#include <avr/pgmspace.h>
#else
#define PROGMEM
#endif
#pragma warning (disable : 4244) // int -> short
#include "bcdmath.h"
#ifndef SMALL_INT
#include "bcd2.h"
#else
typedef BCD BCD2;
#endif
#define BCD_CONST_PI 0
#define BCD_CONST_PI2 1
#define BCD_CONST_PIBY2 2
#define BCD_CONST_PIBY32 3
#define BCD_CONST_SINTAB 4
#define BCD_CONST_COSTAB (BCD_CONST_SINTAB+8)
#define BCD_CONST_LN2 (BCD_CONST_COSTAB+8)
#define BCD_CONST_ONEOTWO (BCD_CONST_LN2+1)
#define BCD_CONST_LN10 (BCD_CONST_ONEOTWO+1)
#define BCD_CONST_LN2OLN10 (BCD_CONST_ONEOTWO+2)
#define BCD_CONST_ATANLIM (BCD_CONST_LN2OLN10+1)
#define BCD_CONST_PIBY32A (BCD_CONST_ATANLIM+1)
#define BCD_CONST_PIBY32B (BCD_CONST_PIBY32A+1)
#define BCD_CONST_HUNDREDTH (BCD_CONST_PIBY32B+1)
#define BCD_CONST_HALF (BCD_CONST_HUNDREDTH+1)
#define BCD_CONST_TWOHALF (BCD_CONST_HALF+1)
#define BCD_CONST_TEN (BCD_CONST_TWOHALF + 1)
#define BCD_CONST_SINPOLY (BCD_CONST_TEN+1)
#define BCD_CONST_COSPOLY (BCD_CONST_SINPOLY+7)
#define BCD_CONST_LOGPOLY (BCD_CONST_COSPOLY+7)
#define BCD_CONST_LOGCK (BCD_CONST_LOGPOLY+4)
#define BCD_CONST_EXP1 (BCD_CONST_LOGCK+64)
#define BCD_CONST_ONEP4 (BCD_CONST_EXP1+1)
#define BCD_CONST_ROOTPI (BCD_CONST_ONEP4+1)
#define BCD_CONST_BCDMIN (BCD_CONST_ROOTPI+1)
#define BCD_CONST_ROOT2 (BCD_CONST_BCDMIN+1)
namespace bcdmath
{
typedef unsigned short Dig[P+1];
#ifdef ARDUINO
#define GET_CONST(_bcd, _c) \
memcpy_P(&_bcd._v, constTable + (_c), sizeof(BCDFloat));
#else
#define GET_CONST(_bcd, _c) \
memcpy(&_bcd._v, constTable + (_c), sizeof(BCDFloat));
#endif
static const Dig constTable[] PROGMEM =
{
{ 3, 1415, 9265, 3589, 7932, 3846, 2643, 1 }, // pi
{ 6, 2831, 8530, 7179, 5864, 7692, 5287, 1 }, // 2pi
{ 1, 5707, 9632, 6794, 8966, 1923, 1322, 1 }, // pi/2
{ 981, 7477, 424, 6810, 3870, 1957, 6057, 0 }, // pi/32
/* table for sin & cos */
{ 980, 1714, 329, 5606, 199, 4195, 5639, 0 }, // sin(pi/32)
{ 1950, 9032, 2016, 1282, 6784, 8284, 8685, 0 }, // sin(pi/16)
{ 2902, 8467, 7254, 4623, 6763, 6192, 3758, 0 }, // sin(3pi/32)
{ 3826, 8343, 2365, 897, 7172, 8459, 9840, 0 }, // sin(pi/8)
{ 4713, 9673, 6825, 9976, 4855, 6387, 6259, 0 }, // sin(5pi/32)
{ 5555, 7023, 3019, 6022, 2474, 2830, 8139, 0 }, // sin(3pi/16)
{ 6343, 9328, 4163, 6454, 9821, 5171, 6132, 0 }, // sin(7pi/32)
{ 7071, 678, 1186, 5475, 2440, 844, 3621, 0 }, // sin(pi/4)
{ 9951, 8472, 6672, 1968, 8624, 4836, 9531, 0 }, // cos(pi/32)
{ 9807, 8528, 403, 2304, 4912, 6182, 2361, 0 }, // cos(pi/16)
{ 9569, 4033, 5732, 2088, 6493, 5797, 8870, 0 }, // cos(3pi/32)
{ 9238, 7953, 2511, 2867, 5612, 8183, 1894, 0 }, // cos(pi/8)
{ 8819, 2126, 4348, 3550, 2971, 2756, 8637, 0 }, // cos(5pi/32)
{ 8314, 6961, 2302, 5452, 3707, 8788, 3776, 0 }, // cos(3pi/16)
{ 7730, 1045, 3362, 7369, 6081, 906, 6098, 0 }, // cos(7pi/32)
{ 7071, 678, 1186, 5475, 2440, 844, 3621, 0 }, // cos(pi/4)
{ 6931, 4718, 559, 9453, 941, 7232, 1215, 0 }, // ln(2)
{ 1, 200, 0, 0, 0, 0, 0, 1 }, // 1.02
{ 2, 3025, 8509, 2994, 456, 8401, 7991, 1 }, // ln(10)
{ 3010, 2999, 5663, 9811, 9521, 3738, 8947, 0 }, // ln(2)/ln(10)
{ 1000, 0, 0, 0, 0, 0, 0, 0 }, // 0.1 atan limit
{ 981, 7477, 424, 0, 0, 0, 0, 0 }, // pi/32 part A
{ 6810, 3870, 1957, 6057, 2748, 4465, 1312, (-3)&EXPMASK }, // pi/32 part B NOTE NEG!!
{ 200, 0, 0, 0, 0, 0, 0, 0 }, // 0.02 limit used in ln1p
{ 5000, 0, 0, 0, 0, 0, 0, 0 }, // 0.5
{ 2, 5000, 0, 0, 0, 0, 0, 1 }, // 2.5
{ 10, 0, 0, 0, 0, 0, 0, 1 }, // 10
// sin(x) polynomial
{ 1666, 6666, 6666, 6666, 6666, 6666, 6365, 32768&EXPMASKNEG },
{ 83, 3333, 3333, 3333, 3333, 3303, 1985, 0 },
{ 1, 9841, 2698, 4126, 9840, 7789, 3780, 32768&EXPMASKNEG },
{ 275, 5731, 9223, 9828, 9145, 4611, 7555, 32767&EXPMASKNEG },
{ 2, 5052, 1083, 7671, 5305, 495, 7411, 65535&EXPMASKNEG },
{ 160, 5903, 801, 9414, 1262, 5005, 7297, 32766&EXPMASKNEG },
{ 7637, 5075, 3356, 6847, 5161, 2933, 1632, 65533&EXPMASKNEG },
// cos(x) polynomial
{ 4999, 9999, 9999, 9999, 9999, 9999, 4857, 32768&EXPMASKNEG },
{ 416, 6666, 6666, 6666, 6666, 6153, 8238, 0 },
{ 13, 8888, 8888, 8888, 8880, 5372, 4713, 32768&EXPMASKNEG },
{ 2480, 1587, 3015, 8220, 374, 6779, 0, 32767&EXPMASKNEG },
{ 27, 5573, 1920, 9146, 3795, 4381, 778, 65535&EXPMASKNEG },
{ 2087, 6734, 8250, 8148, 2157, 8695, 5067, 32766&EXPMASKNEG },
{ 11, 4543, 2933, 7975, 7024, 4861, 9252, 65534&EXPMASKNEG },
// log(x) polynomial
{ 833, 3333, 3333, 3333, 3333, 3200, 4980, 0 },
{ 125, 0, 0, 0, 1767, 3000, 0, 0 },
{ 22, 3214, 2856, 3634, 1290, 0, 0, 0 },
{ 4, 3404, 1799, 7690, 0, 0, 0, 0 },
// log(ck) constants, ck = 1+k/64, k = 1..64;
{ 155, 418, 6535, 9652, 5415, 854, 460, 0 },
{ 307, 7165, 8666, 7536, 8837, 1028, 2075, 0 },
{ 458, 953, 6031, 2942, 316, 6679, 2676, 0 },
{ 606, 2462, 1816, 4348, 4258, 606, 1320, 0 },
{ 752, 2342, 1237, 5875, 2569, 8605, 3399, 0 },
{ 896, 1215, 8689, 6871, 3261, 9951, 4693, 0 },
{ 1037, 9679, 3681, 6435, 6482, 6061, 8037, 0 },
{ 1177, 8303, 5656, 3834, 5453, 8794, 1094, 0 },
{ 1315, 7635, 7788, 7192, 7258, 8716, 1286, 0 },
{ 1451, 8200, 9844, 4978, 9728, 1935, 637, 0 },
{ 1586, 503, 176, 6385, 8409, 3371, 1746, 0 },
{ 1718, 5025, 6926, 6592, 2234, 98, 9460, 0 },
{ 1849, 2233, 8494, 119, 9266, 3903, 5926, 0 },
{ 1978, 2574, 3329, 9198, 8036, 2572, 711, 0 },
{ 2105, 6476, 9107, 3496, 3766, 9552, 8127, 0 },
{ 2231, 4355, 1314, 2097, 5576, 6295, 903, 0 },
{ 2355, 6607, 1312, 7669, 907, 7588, 2189, 0 },
{ 2478, 3616, 3904, 5812, 5678, 602, 7657, 0 },
{ 2599, 5752, 4436, 9260, 6697, 2079, 4945, 0 },
{ 2719, 3371, 5483, 6417, 5883, 1669, 4945, 0 },
{ 2837, 6817, 3130, 6445, 9834, 6901, 2223, 0 },
{ 2954, 6421, 2893, 8358, 7638, 6681, 9060, 0 },
{ 3070, 2503, 5294, 9118, 6207, 5124, 5405, 0 },
{ 3184, 5373, 1118, 5346, 1581, 247, 2135, 0 },
{ 3297, 5328, 6372, 4679, 8181, 4422, 8119, 0 },
{ 3409, 2658, 6970, 5932, 1030, 5089, 1997, 0 },
{ 3519, 7642, 3157, 1781, 8465, 5447, 4562, 0 },
{ 3629, 549, 3689, 3684, 5313, 7824, 3459, 0 },
{ 3737, 1640, 9793, 5840, 8082, 1016, 8327, 0 },
{ 3844, 1169, 8910, 3320, 3973, 4790, 624, 0 },
{ 3949, 9380, 8240, 8689, 7810, 6394, 363, 0 },
{ 4054, 6510, 8108, 1643, 8197, 8013, 1154, 0 },
{ 4158, 2789, 5143, 7109, 6561, 3328, 8929, 0 },
{ 4260, 8439, 5310, 9000, 6312, 4544, 8795, 0 },
{ 4362, 3676, 6774, 9180, 7034, 9041, 3230, 0 },
{ 4462, 8710, 2628, 4195, 1153, 2590, 1806, 0 },
{ 4562, 3743, 3481, 5875, 9438, 805, 5381, 0 },
{ 4660, 8972, 9924, 5992, 2455, 8619, 2475, 0 },
{ 4758, 4590, 4869, 9639, 1426, 5209, 5863, 0 },
{ 4855, 781, 5781, 7008, 780, 1791, 771, 0 },
{ 4950, 7726, 6797, 8515, 1459, 7964, 5848, 0 },
{ 5045, 5601, 752, 3952, 8705, 8308, 5317, 0 },
{ 5139, 4575, 1102, 2343, 1680, 1006, 882, 0 },
{ 5232, 4814, 3764, 5478, 3651, 6807, 2249, 0 },
{ 5324, 6479, 8869, 4718, 4387, 3923, 7234, 0 },
{ 5415, 9728, 2432, 7443, 7157, 6542, 3039, 0 },
{ 5506, 4711, 7952, 6622, 7925, 9948, 1792, 0 },
{ 5596, 1578, 7935, 4226, 8627, 888, 5005, 0 },
{ 5685, 473, 5352, 6687, 1207, 8738, 7648, 0 },
{ 5773, 1536, 5034, 8236, 431, 8112, 615, 0 },
{ 5860, 4904, 5003, 5782, 890, 4119, 4362, 0 },
{ 5947, 710, 7746, 6927, 8951, 4343, 5465, 0 },
{ 6032, 9085, 1438, 842, 6234, 585, 1866, 0 },
{ 6118, 154, 1105, 9929, 352, 9889, 7664, 0 },
{ 6202, 4040, 9751, 8575, 2885, 1494, 6325, 0 },
{ 6286, 865, 9422, 3741, 3774, 4308, 2057, 0 },
{ 6369, 746, 2237, 692, 3162, 494, 4271, 0 },
{ 6451, 3796, 1373, 5847, 166, 5228, 4961, 0 },
{ 6533, 127, 2012, 7456, 3875, 8615, 8812, 0 },
{ 6613, 9848, 2245, 3650, 826, 235, 8387, 0 },
{ 6694, 3065, 3942, 6292, 6729, 8885, 2709, 0 },
{ 6773, 9882, 3591, 8061, 4080, 9682, 6099, 0 },
{ 6853, 400, 3098, 9194, 1654, 4048, 789, 0 },
{ 6931, 4718, 559, 9453, 941, 7232, 1214, 0 },
// e = 2.7182 8182 8459 0452 3536 0287 4713 5266 2497 7572 470936999
{ 2, 7182, 8182, 8459, 452, 3536, 287, 1 }, // exp(1)
{ 1, 4000, 0, 0, 0, 0, 0, 1 }, // 1.4 (used in erf)
{ 1, 7724, 5385, 905, 5160, 2729, 8167, 1 }, // sqrt(PI)
{ 10, 0, 0, 0, 0, 0, 0, 5693 }, // BCD MIN = 1e-9999
{ 1, 4142, 1356, 2373, 950, 4880, 1689, 1 }, // sqrt(2)
};
#ifndef SMALL_INT
#define BCD2_CONST_EXP1 0
#define BCD2_CONST_LANCZOS 1
typedef unsigned short Dig2[P2+1];
static const Dig2 constTable2[] =
{
// 2.71828182845904523536028747135266249775724709369995957496696
{
2, 7182, 8182, 8459, 452, 3536, 287,
4713, 5266, 2497, 7572, 4709, 3699, 9595,7496,
1 },
// double size Lanczos terms for gamma
//3.7948,6229,8882,1576,6137,0571,8385,6087,7748,2487,6086,1116,8266,2325,9950,6316,10
{ 3, 7948, 6229, 8882, 1576, 6137, 571,
8385,6087,7748,2487,6086,1116,8266,2325,
2 },
//-6.1191,9133,3435,0268,9475,3695,8619,2303,8586,6169,7188,5413,0739,6546,3038,1464,6
{ 6, 1191, 9133, 3435, 268, 9475, 3695,
8619,2303,8586,6169,7188,5413,739,6546,
32770 },
//3.1935,3139,9365,7178,9732,6587,2516,1815,7710,7349,5270,6969,8696,7561,8425,94
{ 3, 1935, 3139, 9365, 7178, 9732, 6587,
2516,1815,7710,7349,5270,6969,8696,7562,
2 },
//-1.0648,2204,9659,4145,5971,0636,6767,3045,5636,6427,7798,2642,4908,6556,1951,4320,785
{ 1, 648, 2204, 9659, 4145, 5971, 636,
6767,3045,5636,6427,7798,2642,4908,6556,
32770},
// 2.215,7659,2545,9700,1065,2640,0839,0578,0196,6530,8981,3632,5470,7872,885
{ 2215, 7659, 2545, 9700, 1065, 2640, 839,
578,196,6530,8981,3632,5470,7872,885,
1 },
//-2.77,3434,9002,3102,0315,6808,5633,2632,6001,3314,1546,7408,3072,4481,5716,8215,697
{ 277, 3434, 9002, 3102, 315, 6808, 5633,
2632,6001,3314,1546,7408,3072,4481,5717,
32769 },
//1.9,7504,4798,8896,0954,2464,2454,3527,7206,3456,0350,8210,3826,3387,9892,0783,963
{ 19, 7504, 4798, 8896, 954, 2464, 2454,
3527,7206,3456,350,8210,3826,3387,9892,
1 },
// -7.351,4584,5326,3110,3427,1541,0972,9593,4702,3887,1286,0444,6989,7138,7333,92
{ 7351, 4584, 5326, 3110, 3427, 1541, 972,
9593,4702,3887,1286,444,6989,7138,7334,
32768 },
// 1.25,0201,6315,9372,8926,0576,1395,0748,8131,6554,2830,4183,3646,7890,817
{ 125, 201, 6315, 9372, 8926, 576, 1395,
748,8131,6554,2830,4183,3646,7890,817,
0 },
// -7.710,2871,8096,9904,7327,0526,2403,7596,8997,0952,8489,4696,4202,3767,5221,521
{ 7710, 2871, 8096, 9904, 7327, 526, 2403,
7596,8997,952,8489,4696,4202,3767,5222,
40959 },
//1.0,9373,7115,9701,7175,1506,3503,3523,6971,8098,6730,4848,3630,9464,6226,8883,4744,5305
{ 10, 9373, 7115, 9701, 7175, 1506, 3503,
3523,6971,8098,6730,4848,3630,9464,6227,
8191 },
//-1.1,2406,1022,3182,8735,6453,7307,4629,1188,2930,0182,3911,2707,2798,2287,6267,78
{ 11, 2406, 1022, 3182, 8735, 6453, 7307,
4629,1188,2930,182,3911,2707,2798,2288,
40958 },
//-2.77,0957,5972,2463,9573,5873,7503,6522,6385,2612,8760,1919,8791,4529,9276,1905,83
{ 277,957,5972,2463,9573,5873,7503,
6522,6385,2612,8760,1919,8791,4529,9276,
40956 },
};
#endif // !SMALL_INT
BCD pi()
{
BCD v;
GET_CONST(v, BCD_CONST_PI);
return v;
}
// 2pi
BCD pi2()
{
BCD v;
GET_CONST(v, BCD_CONST_PI2);
return v;
}
static void horner(int cbase, BCD& x, int n, BCD& res)
{
int i;
int cn = cbase + n;
res = 0;
for (i = 0; i < n; ++i)
{
res *= x;
BCD c; --cn; GET_CONST(c, cn);
res += c;
}
}
static void sinPoly(const BCD& a, BCD& sa)
{
/* calculate sin(a) for |a| <= pi/32
*
* p(y) =
-0.166666666666666666666666636459634636681163368103
+ 0.00833333333333333333330319847154774103100919421 *y
- 0.000198412698412698407789377962065944968466530592 * y**2
+ 2.755731922398289145461175553432349911142734e-6 * y**3
- 2.5052108376715305049574108322138191235704e-8 *y**4
+ 1.60590308019414126250057297007175565961045303085e-10 *y**5
- 7.63750753356684751612933163194608455774099589e-13 *y**6
* then sin(x) ~ x(1+x^2*p(x^2)). relative error <= 1.25*10^-28.
*
* NOTES:
* MiniMaxApproximation[
If[y > 0, (Sin[Sqrt[y]] - Sqrt[y])/y^(3/2), -1/6], {y, {0, Pi/32}, 6, 0},
WorkingPrecision -> 30][[2, 1]]
*
*/
BCD y = a*a;
BCD py;
horner(BCD_CONST_SINPOLY, y, 7, py);
sa = a*(1+y*py);
}
static void cosPoly(const BCD& a, BCD& ca)
{
/*
* calculate sin(a) for |a| <= pi/32
*
* p(y) =
-0.499999999999999999999999485707905851100423176682
+ 0.041666666666666666666153823812611360817535526463*y
-0.001388888888888888805372471283461577523825976975*Power(y,2)
+ 0.000024801587301582200374677934077121988251491037*Power(y,3)
- 2.75573192091463795438107775847721847637672e-7*Power(y,4)
+ 2.087673482508148215786955066969762215737996819883e-9*Power(y,5)
- 1.1454329337975702448619252226154423156415441444e-11*Power(y,6)
*
* then cos(x) + 1+x^2*p(x^2). relative error 3*10^-26.
*
* NOTES:
MiniMaxApproximation[
If[y > 0, (Cos[Sqrt[y]] - 1)/y, -1/2], {y, {0, Pi/32}, 6, 0},
WorkingPrecision -> 30][[2, 1]]
*
*/
BCD y = a*a;
BCD py;
horner(BCD_CONST_COSPOLY, y, 7, py);
ca = 1+y*py;
}
#define K 12
#define GG 12
#if 0
#include <stdio.h>
static void dump(const BCD& v)
{
int i;
printf("{ ");
for (i = 0; i < P; ++i)
printf("%d, ", v._v.d_[i]);
printf("%d },\n", v._v.d_[P]);
}
#endif
static BCD _expm1(const BCD& a)
{
BCD t(1);
int d = 2;
BCD2 s(1);
BCD2 s1;
BCD ss;
for (;;)
{
t *= a/d;
s1 = s + t;
ss = s1.asBCD();
if (ss == s.asBCD())
{
ss *= a;
break;
}
s = s1;
++d;
}
return ss;
}
#if 0
/* This is not the version to use. it is the version that calculates
* the coefficients as it goes. It is used to recalculate the constants
* if we change K or GG.
*/
BCD _gammaFactorialSlow(const BCD& z)
{
/* calculate gamma(z+1) = z!
* using lanczos expansion.
*/
BCD t1;
BCD t2;
BCD s;
BCD t;
BCD fgc[K+1];
BCD pks;
BCD aa;
int a[K];
int b[K+1];
int tb;
int i, j;
BCD half; GET_CONST(half, BCD_CONST_HALF);
a[0] = 1;
b[0] = -1;
b[1] = 2;
fgc[0] = exp(GG+half)/sqrt(GG+half);
s = fgc[0]*half;
//print(s); // emit coefficient
aa = half;
t = 1;
i = 1;
for (;;)
{
t *= (z+(1-i))/(z + i);
t1 = i + half;
t2 = t1 + GG;
fgc[i] = aa*exp(t2 - log(t2)*t1);
aa *= t1;
pks = 0;
for (j = 0; j <= i; ++j)
pks += b[j]*fgc[j];
//print(pks); // emit coefficient
s += pks*t;
if (i == K) break;
a[i] = 0;
b[i+1] = 0;
tb = b[0];
for (j = 0; j <= i; ++j)
{
a[j] = -a[j] + 2*tb;
tb = b[j+1];
b[j+1] = -b[j+1] + 2*a[j];
}
b[0] = -b[0];
++i;
}
t1 = z + half;
t2 = t1 + GG;
return 2*exp(t1*log(t2)-t2)*s;
}
#endif
void sincos(const BCD& v, BCD* sinv, BCD* cosv)
{
/* calculate sin(v), cos(v) or both as requested.
*/
BCD a;
int k;
bool neg = v.isNeg();
/* arrange a >= 0 */
a = v;
a._v.clearSign();
/* reduce argument to 0 <= a < 2pi using special means,
* taking care of large arguments (eg sin(1e22)). its possible
* that the answer cannot be calculated accurately, in which case
* modtwopi bails out with NAN.
*/
a = modtwopi(a);
if (a.isSpecial())
{
if (sinv) *sinv = a;
if (cosv) *cosv = a;
return;
}
/*
* reduce to k*pi/32 + a, where a < pi/32. use a lookup table
* for sin(k*pi/32) and cos(k*pi/32). require 8 entries for each.
*/
BCD piby32; GET_CONST(piby32, BCD_CONST_PIBY32);
BCD a32 = a/piby32;
k = ifloor(a32);
if (k > 0)
{
// subtract in two parts for accuracy.
BCD piby32a; GET_CONST(piby32a, BCD_CONST_PIBY32A);
BCD piby32b; GET_CONST(piby32b, BCD_CONST_PIBY32B);
//a = a - k*piby32a - k*piby32b;
BCD kf((unsigned int)k);
a -= kf*piby32a;
a -= kf*piby32b;
}
#ifdef TRIG_SPECIAL_CASES
// special case tests
// numbers very close to special values
#define IS_SMALL(_v) ((_v).isZero() || (_v).exponent() <= -5)
int k1 = k;
// are we npi/32 + eps;
bool npi32 = IS_SMALL(a);
if (!npi32)
{
// are we npi/32 - eps
BCD b = a - piby32;
npi32 = IS_SMALL(b);
++k1;
}
if (npi32)
{
if ((k1 & 15) == 0)
{
k1 >>= 4;
bool odd = k1 & 1;
if (sinv)
{
if (!odd) *sinv = 0;
else
{
*sinv = 1;
if (k1 == 3) sinv->negate();
}
}
if (cosv)
{
if (odd) *cosv = 0;
else
{
*cosv = 1;
if (k1 == 2) cosv->negate();
}
}
return;
}
}
#endif // TRIG_SPECIAL_CASES
/* now a <= pi/32, we use the polynomial approximations */
BCD sa;
BCD ca;
BCD sina, cosa;
BCD* sap = 0;
BCD* cap = 0;
k &= 64-1; // wrap around at 2pi
int q = (k>>4); // q is the quadrant.
k &= 15; // index into table.
if (k) // sap & cap = 0 for k = 0.
{
/* require both sin and cos */
sinPoly(a, sa);
cosPoly(a, ca);
BCD sk;
BCD ck;
if (k < 8)
{
GET_CONST(sk, BCD_CONST_SINTAB-1+k);
GET_CONST(ck, BCD_CONST_COSTAB-1+k);
//sina = sk*ca + ck*sa;
//cosa = ck*ca - sk*sa;
// these version used less stack under gcc
sina = sk*ca;
sina += ck*sa;
cosa = ck*ca;
cosa -= sk*sa;
}
else
{
GET_CONST(sk, BCD_CONST_SINTAB+15-k);
GET_CONST(ck, BCD_CONST_COSTAB+15-k);
//sina = ck*ca + sk*sa;
//cosa = sk*ca - ck*sa;
sina = ck*ca;
sina += sk*sa;
cosa = sk*ca;
cosa -= ck*sa;
}
sap = &sina;
cap = &cosa;
}
if (sinv)
{
if ((q&1) == 0)
{
if (!sap) { sinPoly(a, sina); sap = &sina; }
*sinv = sina;
}
else
{
if (!cap) { cosPoly(a, cosa); cap = &cosa; }
*sinv = cosa;
}
if (neg ^ (q > 1)) sinv->negate(); // *sinv = -*sinv;
}
if (cosv)
{
if ((q&1) == 0)
{
if (!cap) cosPoly(a, *cosv);
else *cosv = cosa;
}
else
{
if (!sap) sinPoly(a, *cosv);
else *cosv = sina;
}
if ((q == 1) | (q == 2)) cosv->negate(); // *cosv = -*cosv;
}
}
void sinhcosh(const BCD& a, BCD* sinha, BCD* cosha)
{
BCD v = exp(a);
if (sinha) *sinha = (v - 1/v)/2; // XX unstable
if (cosha) *cosha = (v + 1/v)/2;
}
BCD sin(const BCD& v)
{
BCD sinv;
if (v.isSpecial()) return v;
sincos(v, &sinv, 0);
return sinv;
}
BCD sinDeg(const BCD& v)
{
BCD sinv;
if (v.isSpecial()) return v;
BCD a(360);
BCD v1 = fmod(v, a)*pi2()/a;
sincos(v1, &sinv, 0);
return sinv;
}
BCD sinGrad(const BCD& v)
{
BCD sinv;
if (v.isSpecial()) return v;
BCD a(400);
BCD v1 = fmod(v, a)*pi2()/a;
sincos(v1, &sinv, 0);
return sinv;
}
BCD cos(const BCD& v)
{
BCD cosv;
if (v.isSpecial()) return v;
sincos(v, 0, &cosv);
return cosv;
}
BCD cosDeg(const BCD& v)
{
BCD cosv;
if (v.isSpecial()) return v;
BCD a(360);
BCD v1 = fmod(v, a)*pi2()/a;
sincos(v1, 0, &cosv);
return cosv;
}
BCD cosGrad(const BCD& v)
{
BCD cosv;
if (v.isSpecial()) return v;
BCD a(400);
BCD v1 = fmod(v, a)*pi2()/a;
sincos(v1, 0, &cosv);
return cosv;
}
BCD tanDeg(const BCD& v)
{
BCD s, c;
if (v.isSpecial()) return v;
BCD a(360);
BCD v1 = fmod(v, a)*pi2()/a;
sincos(v1, &s, &c);
return s/c;
}
BCD tanGrad(const BCD& v)
{
BCD s, c;
if (v.isSpecial()) return v;
BCD a(400);
BCD v1 = fmod(v, a)*pi2()/a;
sincos(v1, &s, &c);
return s/c;
}
BCD tan(const BCD& v)
{
BCD s, c;
if (v.isSpecial()) return v;
sincos(v, &s, &c);
return s/c;
}
BCD powfmod(const BCD& a, int4 n, const BCD& fm)
{
// ASSUME n >= 0
// ASSUME a >= 0, fm > 0
if (n == 0) return 1;
BCD s;
BCD r = a;
s = 1;
/* Use binary exponentiation */
for (;;)
{
if (n & 1)
{
s *= r;
if (s >= fm)
{
BCD q = trunc(s/fm);
s -= q*fm;
}
}
n >>= 1;
if (!n) break;
r *= r;
if (r >= fm)
{
BCD q = trunc(r/fm);
r -= q*fm;
}
}
return s;
}
BCD exp(const BCD& v)
{
/* write v = k*r + n*ln(2)
* where |k*r| < ln(2)
*
* let k be some power of 2, eg 64 so that
*
* exp(v) = exp(kr + nln(2)) = exp(kr)*2^n
* = exp(r)^64 * 2^n
*
* then r is small enough for Taylor series.
*/
if (v.isSpecial())
{
if (v.isInf() && v.isNeg()) return 0; // exp(-inf) = 0
return v;
}
if (v.isZero()) return 1;
BCD ln2; GET_CONST(ln2, BCD_CONST_LN2);
BCD n = trunc(v/ln2);
if (n > 33218)
{
/* overflow */
return BCDFloat::posInf();
}
else if (n < -33218)
{
/* underflow */
return 0;
}
int ni = itrunc(n);
int k = 64;
BCD r = (v - n*ln2)/k;
BCD t1 = _expm1(r);
BCD er = t1 + 1;
BCD del = t1 - (er - 1);
t1 = pow(er,k);
er = t1 + t1/er*del*k; // correction factor
return er*pow(BCD(2), ni);
}
static BCD _ln1p(const BCD& a)
{
/* otherwise use a series that converges for small arguments */
BCD s;
BCD t;
BCD s1;
BCD x;
BCD c;
c = 2;
s = a;
x = -a;
t = x;
for (;;)
{
t *= x;
s1 = s - t/c;
if (s1 == s) break;
c += 1;
s = s1;
}
return s1;
}
static BCD logPoly(const BCD& r)
{
/*
MiniMaxApproximation[
If[r > 0, Log[(1 + Sqrt[r]/2)/(1 - Sqrt[r]/2)]/Sqrt[r],
1], {r, {0, 1/128}, 3, 0}, WorkingPrecision -> 30]
produces p(r), so ln((1+r/2)/(1-r/2)) ~ p(r^2)*r
BUT, it's not very good, seeing about |e| ~ 1.25e-14
*/
/* p(r) =
r + 0.08333333333333335805 r^3 + 0.0124999999978878 r^5 +
0.002232197165 r^7
max error <= 0.27e-24.
Elementary Algorithms. Jean-Michel Muller. p57.
*/
/*
p(r) =
r + 0.0833333333333333333332004980000 r^3 +
0.0125000000000000176730000000000 r^5 +
0.00223214285636341290000000000000 r^7 +
0.000434041799769000000000000000000 r^9
max error <= 0.80e-30.
Elementary Algorithms. Jean-Michel Muller. p57.
relative error max of 3e-28.
*/
BCD r2 = r*r;
BCD pr;
horner(BCD_CONST_LOGPOLY, r2, 4, pr);
return (pr*r2 + 1)*r;
}
static BCD _log(const BCD& v, int4& p10, int4& p2, bool& neg)
{
if (v.isNeg()) return BCDFloat::nan();
if (v.isSpecial()) return v;
if (v.isZero()) return BCDFloat::negInf();
BCD lna;
BCD2 a2;
neg = false;
if (v < 1)
{
// v in (0,1) map to (1,inf) and negate final answer
a2 = BCD2(1U)/v;
neg = true;
if (a2.isInf())
{
// This only happens if v = 1e-10000
p10 = 10000;
p2 = 0;
return 0;
}
}
else
a2 = v;
// have a in [1,inf)
// extract the power of 10 from the exponent
p10 = (a2.exponent()-1)*4;
a2.setExponent(1);
// divide out the biggest digit to leave a number < 10
unsigned int d = a2.digit(0);
int dnorm = 1;
while (d >= 10)
{
++p10; // adjust the 10's count
dnorm *= 10;
d /= 10;
}
// divide by 2 until < 2, keep a count of the 2's
p2 = 0;
while (d > 1)
{
++p2; // adjust the 2s count
dnorm <<= 1;
d >>= 1;
}
// finally divide the original number by the dnorm to take out
// the 10s and 2s, leaving a value not bigger than 2.
if (dnorm > 1)
a2 /= dnorm;
// we have a in (1+delta,2]
// ln(v) = ln(a) + p10*ln10 + p2*ln2;
// find ck where ck = 1+k/64, k=1,2,..64 with |x-ck| <= 1/128
BCD2 a21 = a2 - 1;
BCD half; GET_CONST(half, BCD_CONST_HALF);
int k = ifloor(a21.asBCD()*64 + half); // round to nearest k
// set r = 2*(a - ck)/(a + ck)
BCD2 dk = BCD(k)/64;