Eigen 大大好聰明哦，羨慕羨慕......

請問一下，我用計算機按 tan(90度) 為何會出現 Error，但 89 度卻有數值，超過 90 度後卻變成負數，為什麼會這樣子？

還有一個問題，就是 C 語言內建的函數數庫是不是因為要能夠計算到 360 度的全域範圍，而且有負號的情況，所以才需要複雜的浮點數運算，謝謝。

那是查表的的第幾個




這是用 excel 建的table

先在 A 欄 建立 1~255
b 欄 =a1/256*pi()
c 欄 =tan(b1)
d 欄 =c1*256
e 欄 =dec2hex(d1,4)

這樣我們就將 -127/128π ~ 127/128π 的 tan 值建立出來

之後只要反過來找就可以了

將你的數值*256 後，和 e 欄找出最接近的值，然後對應到 a 就是 arctan了




我剛剛又想到可以查得更精準
0~1/2π 的值和 0~-1/2π 的值事實上是一樣的，只是差個正負號罷了

所以我只要建立 0~90℃ 的表就可以了

所以我用 excel

先在 A 欄 建立 0~179
b 欄 =tan(a1/2*pi())*512
c 欄 =dec2hex(b1,4)

然後將 C欄 的資料整理下來

0x0000,0x0004,0x0008,0x000D,0x0011,0x0016,0x001A,0x001F,
0x0023,0x0028,0x002C,0x0031,0x0035,0x003A,0x003E,0x0043,;0

0x0047,0x004C,0x0051,0x0055,0x005A,0x005E,0x0063,0x0068,
0x006C,0x0071,0x0076,0x007A,0x007F,0x0084,0x0089,0x008D,;1

0x0092,0x0097,0x009C,0x00A1,0x00A6,0x00AB,0x00B0,0x00B5,
0x00BA,0x00BF,0x00C4,0x00C9,0x00CE,0x00D4,0x00D9,0x00DE,

0x00E3,0x00E9,0x00EE,0x00F4,0x00F9,0x00FF,0x0104,0x010A,
0x0110,0x0115,0x011B,0x0121,0x0127,0x012D,0x0133,0x0139,

0x013F,0x0146,0x014C,0x0152,0x0159,0x015F,0x0166,0x016D,
0x0173,0x017A,0x0181,0x0188,0x0190,0x0197,0x019E,0x01A6,

0x01AD,0x01B5,0x01BD,0x01C4,0x01CD,0x01D5,0x01DD,0x01E5,
0x01EE,0x01F7,0x01FF,0x0209,0x0212,0x021B,0x0225,0x022E,

0x0238,0x0242,0x024C,0x0257,0x0262,0x026D,0x0278,0x0283,
0x028F,0x029B,0x02A7,0x02B3,0x02C0,0x02CD,0x02DB,0x02E8,

0x02F7,0x0305,0x0314,0x0323,0x0333,0x0343,0x0354,0x0365,
0x0376,0x0388,0x039B,0x03AE,0x03C2,0x03D7,0x03EC,0x0402,

0x0419,0x0431,0x0449,0x0463,0x047D,0x0499,0x04B6,0x04D4,
0x04F3,0x0513,0x0535,0x0559,0x057E,0x05A5,0x05CE,0x05FA,

0x0627,0x0657,0x068A,0x06C0,0x06F9,0x0736,0x0776,0x07BB,
0x0805,0x0854,0x08A9,0x0905,0x0968,0x09D4,0x0A4A,0x0ACA,

0x0B57,0x0BF3,0x0CA0,0x0D61,0x0E3B,0x0F31,0x1049,0x118D,
0x1307,0x14C5,0x16DC,0x1969,0x1C99,0x20B3,0x2629,0x2DCE,

0x3945,0x4C60,0x7294,0xE52D,

現在我們要棤 0.255 ，只要 *512=0x83 再去查表，

我們會發現資料會落在 0x007F(0x1c),0x0084(0x1d)
數值比較接近 0x1d =29 *0.5℃=14.5℃

tan(14.5℃)=0.25861758435589028187700082104918





更簡單方便的查表

先在 A 欄 建立 0~179
b 欄 =tan((a1+0.25)/2*pi())*1024
c 欄 =dec2hex(b1,4)

之後你發現 數直只能算到 88.5℃

然後你只要查表一，找到最接近的，就是了，不用再管是靠近上值，還是靠近下值

0x0004,0x000D,0x0016,0x001F,0x0028,0x0031,0x003A,0x0043,
0x004C,0x0055,0x005E,0x0067,0x0070,0x0079,0x0082,0x008B,

0x0094,0x009D,0x00A6,0x00AF,0x00B9,0x00C2,0x00CB,0x00D4,
0x00DE,0x00E7,0x00F1,0x00FA,0x0104,0x010D,0x0117,0x0120,

0x012A,0x0134,0x013D,0x0147,0x0151,0x015B,0x0165,0x016F,
0x0179,0x0183,0x018E,0x0198,0x01A2,0x01AD,0x01B7,0x01C2,

0x01CD,0x01D8,0x01E2,0x01ED,0x01F8,0x0204,0x020F,0x021A,
0x0226,0x0231,0x023D,0x0249,0x0255,0x0261,0x026D,0x0279,

0x0286,0x0292,0x029F,0x02AC,0x02B9,0x02C6,0x02D3,0x02E1,
0x02EE,0x02FC,0x030A,0x0318,0x0327,0x0335,0x0344,0x0353,

0x0362,0x0372,0x0382,0x0391,0x03A2,0x03B2,0x03C3,0x03D4,
0x03E5,0x03F7,0x0408,0x041B,0x042D,0x0440,0x0453,0x0467,

0x047B,0x048F,0x04A4,0x04B9,0x04CF,0x04E5,0x04FB,0x0512,
0x052A,0x0542,0x055B,0x0574,0x058E,0x05A8,0x05C4,0x05DF,

0x05FC,0x0619,0x0637,0x0656,0x0676,0x0697,0x06B9,0x06DB,
0x06FF,0x0724,0x074A,0x0771,0x079A,0x07C4,0x07EF,0x081C,

0x084A,0x087B,0x08AD,0x08E1,0x0917,0x094F,0x0989,0x09C6,
0x0A06,0x0A49,0x0A8E,0x0AD7,0x0B24,0x0B74,0x0BC8,0x0C21,

0x0C7E,0x0CE1,0x0D4A,0x0DB9,0x0E2E,0x0EAB,0x0F31,0x0FC0,
0x1058,0x10FC,0x11AD,0x126C,0x133B,0x141B,0x1511,0x161E,

0x1747,0x188F,0x19FD,0x1B96,0x1D64,0x1F71,0x21CB,0x2486,
0x27B9,0x2B88,0x3023,0x35D3,0x3D07,0x4671,0x5346,0x65CE,

0x82EB,0xB751,

同樣的 查 0.255 *1024 =261= 0x0105 <0x010D(0x1d)

0x1d *0.5 =14.5℃

請問 Eigen 大大：

arctan(0.255) 0.255*2048 =0x020a ,比較到 0x0201(0x8e),0x021C(0x8f),

0x0201 為什麼等於 (0x8e)
0x021C 為什麼等於 (0x8f)

請問怎麼來的？謝謝。

我是用泰勒方式計算, 誤差很小.

tab: -127/256~127/256 pi
0xAE84,0xD744,0xE4DA,0xEBA5,0xEFB9,0xF271,0xF463,0xF5D9,0xF6FC,0xF7E4,0xF8A3,0xF942,0xF9C9,0xFA3D,0xFAA1,0xFAF9,
0xFB47,0xFB8C,0xFBCA,0xFC02,0xFC35,0xFC63,0xFC8D,0xFCB4,0xFCD8,0xFCF9,0xFD18,0xFD35,0xFD4F,0xFD68,0xFD80,0xFD96,
0xFDAB,0xFDBE,0xFDD1,0xFDE3,0xFDF3,0xFE03,0xFE13,0xFE21,0xFE2F,0xFE3C,0xFE49,0xFE55,0xFE61,0xFE6C,0xFE76,0xFE81,
0xFE8B,0xFE95,0xFE9E,0xFEA7,0xFEB0,0xFEB8,0xFEC0,0xFEC8,0xFED0,0xFED7,0xFEDE,0xFEE6,0xFEEC,0xFEF3,0xFEFA,0xFF00,
0xFF06,0xFF0C,0xFF12,0xFF18,0xFF1E,0xFF23,0xFF29,0xFF2E,0xFF33,0xFF38,0xFF3D,0xFF42,0xFF47,0xFF4C,0xFF50,0xFF55,
0xFF59,0xFF5E,0xFF62,0xFF67,0xFF6B,0xFF6F,0xFF73,0xFF77,0xFF7B,0xFF7F,0xFF83,0xFF87,0xFF8B,0xFF8F,0xFF92,0xFF96,
0xFF9A,0xFF9D,0xFFA1,0xFFA4,0xFFA8,0xFFAB,0xFFAF,0xFFB2,0xFFB6,0xFFB9,0xFFBD,0xFFC0,0xFFC3,0xFFC7,0xFFCA,0xFFCD,
0xFFD0,0xFFD4,0xFFD7,0xFFDA,0xFFDD,0xFFE0,0xFFE4,0xFFE7,0xFFEA,0xFFED,0xFFF0,0xFFF3,0xFFF7,0xFFFA,0xFFFD,0x0000,
0x0003,0x0006,0x0009,0x000D,0x0010,0x0013,0x0016,0x0019,0x001C,0x0020,0x0023,0x0026,0x0029,0x002C,0x0030,0x0033,
0x0036,0x0039,0x003D,0x0040,0x0043,0x0047,0x004A,0x004E,0x0051,0x0055,0x0058,0x005C,0x005F,0x0063,0x0066,0x006A,
0x006E,0x0071,0x0075,0x0079,0x007D,0x0081,0x0085,0x0089,0x008D,0x0091,0x0095,0x0099,0x009E,0x00A2,0x00A7,0x00AB,
0x00B0,0x00B4,0x00B9,0x00BE,0x00C3,0x00C8,0x00CD,0x00D2,0x00D7,0x00DD,0x00E2,0x00E8,0x00EE,0x00F4,0x00FA,0x0100,
0x0106,0x010D,0x0114,0x011A,0x0122,0x0129,0x0130,0x0138,0x0140,0x0148,0x0150,0x0159,0x0162,0x016B,0x0175,0x017F,
0x018A,0x0194,0x019F,0x01AB,0x01B7,0x01C4,0x01D1,0x01DF,0x01ED,0x01FD,0x020D,0x021D,0x022F,0x0242,0x0255,0x026A,
0x0280,0x0298,0x02B1,0x02CB,0x02E8,0x0307,0x0328,0x034C,0x0373,0x039D,0x03CB,0x03FE,0x0436,0x0474,0x04B9,0x0507,
0x055F,0x05C3,0x0637,0x06BE,0x075D,0x081C,0x0904,0x0A27,0x0B9D,0x0D8F,0x1047,0x145B,0x1B26,0x28BC,0x517C,


TAB_H,TAB_L 組合成一個 16bit帶正負號的資料

將你的 數值，*256 之後, 和 TAB_H,TAB_L 做比較

ex: arctan(0.255) 0.255*256 =0x0041 ,比較到 0x0040 (0x93),0x0041,(0x94)

比較接近 0x93

所以arctan(0.255) = ((0x93+1)-128)/256 pi =20/256 pi =0.078125 pi

tan(0.0.078125 pi )= 0.25048696019130546159570216012472

大概是這樣做，主要的限制是tan是發散的，因此在 -90 及 90 會變成無限大，使得整個查表精度難以控制

上表以是 tan( 127/128 pin ) =82 下去建的

如果將角度限制在 -86~ 86 ，那查表值的精度會變成

-122/256~122/256 pi

0x938C,0xA31A,0xAEC6,0xB7DD,0xBF23,0xC519,0xCA12,0xCE48, ;0
0xD1E5,0xD508,0xD7C8,0xDA36,0xDC60,0xDE50,0xE010,0xE1A6,
0xE317,0xE46A,0xE5A1,0xE6BF,0xE7C9,0xE8BF,0xE9A4,0xEA7A, ;1
0xEB43,0xEBFF,0xECB0,0xED56,0xEDF4,0xEE89,0xEF16,0xEF9C,
0xF01B,0xF095,0xF108,0xF177,0xF1E1,0xF246,0xF2A7,0xF304, ;2
0xF35E,0xF3B4,0xF407,0xF457,0xF4A4,0xF4EF,0xF537,0xF57C,
0xF5C0,0xF601,0xF641,0xF67E,0xF6BA,0xF6F4,0xF72C,0xF763, ;3
0xF799,0xF7CD,0xF800,0xF832,0xF862,0xF892,0xF8C0,0xF8ED,
0xF919,0xF945,0xF96F,0xF999,0xF9C2,0xF9EA,0xFA11,0xFA38, ;4
0xFA5E,0xFA83,0xFAA8,0xFACC,0xFAEF,0xFB12,0xFB34,0xFB56,
0xFB78,0xFB99,0xFBB9,0xFBD9,0xFBF9,0xFC18,0xFC37,0xFC56, ;5
0xFC74,0xFC92,0xFCB0,0xFCCD,0xFCEA,0xFD07,0xFD23,0xFD3F,
0xFD5B,0xFD77,0xFD93,0xFDAE,0xFDC9,0xFDE4,0xFDFF,0xFE1A, ;6
0xFE34,0xFE4E,0xFE69,0xFE83,0xFE9D,0xFEB6,0xFED0,0xFEEA,
0xFF03,0xFF1D,0xFF36,0xFF50,0xFF69,0xFF82,0xFF9B,0xFFB5, ;7
0xFFCE,0xFFE7,0x0000,0x0019,0x0032,0x004B,0x0065,0x007E,
0x0097,0x00B0,0x00CA,0x00E3,0x00FD,0x0116,0x0130,0x014A, ;8
0x0163,0x017D,0x0197,0x01B2,0x01CC,0x01E6,0x0201,0x021C,
0x0237,0x0252,0x026D,0x0289,0x02A5,0x02C1,0x02DD,0x02F9, ;9
0x0316,0x0333,0x0350,0x036E,0x038C,0x03AA,0x03C9,0x03E8,
0x0407,0x0427,0x0447,0x0467,0x0488,0x04AA,0x04CC,0x04EE,
0x0511,0x0534,0x0558,0x057D,0x05A2,0x05C8,0x05EF,0x0616,
0x063E,0x0667,0x0691,0x06BB,0x06E7,0x0713,0x0740,0x076E,
0x079E,0x07CE,0x0800,0x0833,0x0867,0x089D,0x08D4,0x090C,
0x0946,0x0982,0x09BF,0x09FF,0x0A40,0x0A84,0x0AC9,0x0B11,
0x0B5C,0x0BA9,0x0BF9,0x0C4C,0x0CA2,0x0CFC,0x0D59,0x0DBA,
0x0E1F,0x0E89,0x0EF8,0x0F6B,0x0FE5,0x1064,0x10EA,0x1177,
0x120C,0x12AA,0x1350,0x1401,0x14BD,0x1586,0x165C,0x1741,
0x1837,0x1941,0x1A5F,0x1B96,0x1CE9,0x1E5A,0x1FF0,0x21B0,
0x23A0,0x25CA,0x2838,0x2AF8,0x2E1B,0x31B8,0x35EE,0x3AE7,
0x40DD,0x4823,0x513A,0x5CE6,0x6C74,0x8237,


ex: arctan(0.255) 0.255*2048 =0x020a ,比較到 0x0201(0x8e),0x021C(0x8f),

比較接近 0x8e

所以arctan(0.255) = ((0x8e+6)-128)/256 pi =20/256 pi =0.078125 pi

tan(0.0.078125 pi )= 0.25048696019130546159570216012472

我需要的是 arc tan 反三角函數，請問有辦法用查表的方式嗎？

三角反三角
用建表查表方式不是也可以嗎?
還是您需要比較精確結果

大大們好：
請問是否能用C語言以四則運算實現反三角函數運算，，並且不用內建的函數庫般的使用浮點數運算，因為 16F 系列的 PIC 堆疊實在不夠用，編譯的時候常常會OVERFLOW，所以想找看看有無以手算的方式或是範例程式可用。

謝謝。