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Replace sine table with 5th order approximation
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The previous implementation used the pre-calculated fixed-point
arithmetic table for sine values, which added over 2 KiB to the code
size.

To solve this issue, I implemented the 5th order polynomial
approximation for sine[1], which reduced the code size from 2556 bytes
to 918 bytes (by about 65%) while maintaining acceptable error (RMS
error is 8.866 observed with 0-90 degrees, 0-1024 in fixed-point
representation)

Besides, as several applications need the sine and cosine of the same
angle, I replace all occurrences of twin_sin and twin_cos with the
shortcut twin_sincos, including the code in the files src/path.c and
src/matrix.c.

[1]: https://www.nullhardware.com/blog/fixed-point-sine-and-cosine-for-embedded-systems/

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@ndsl7109256
ndsl7109256 committed Aug 9, 2024
1 parent db1162e commit ef31d46
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Showing 4 changed files with 83 additions and 153 deletions.
2 changes: 2 additions & 0 deletions include/twin.h
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Original file line number Diff line number Diff line change
Expand Up @@ -1042,6 +1042,8 @@ twin_fixed_t twin_cos(twin_angle_t a);

twin_fixed_t twin_tan(twin_angle_t a);

void twin_sincos(twin_angle_t a, twin_fixed_t *sin, twin_fixed_t *cos);

/*
* widget.c
*/
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4 changes: 2 additions & 2 deletions src/matrix.c
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Original file line number Diff line number Diff line change
Expand Up @@ -100,8 +100,8 @@ twin_point_t _twin_matrix_expand(twin_matrix_t *matrix)
void twin_matrix_rotate(twin_matrix_t *m, twin_angle_t a)
{
twin_matrix_t t;
twin_fixed_t c = twin_cos(a);
twin_fixed_t s = twin_sin(a);
twin_fixed_t c, s;
twin_sincos(a, &s, &c);

t.m[0][0] = c;
t.m[0][1] = s;
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14 changes: 10 additions & 4 deletions src/path.c
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Original file line number Diff line number Diff line change
Expand Up @@ -134,6 +134,13 @@ void twin_path_draw(twin_path_t *path, twin_fixed_t x, twin_fixed_t y)
_twin_matrix_y(&path->state.matrix, x, y));
}

static void twin_path_draw_polar(twin_path_t *path, twin_angle_t deg)
{
twin_fixed_t s, c;
twin_sincos(deg, &s, &c);
twin_path_draw(path, c, s);
}

void twin_path_rdraw(twin_path_t *path, twin_fixed_t dx, twin_fixed_t dy)
{
twin_spoint_t here = _twin_path_current_spoint(path);
Expand Down Expand Up @@ -218,14 +225,13 @@ void twin_path_arc(twin_path_t *path,
twin_angle_t last = (start + extent - inc + epsilon) & ~(step - 1);

if (first != start)
twin_path_draw(path, twin_cos(start), twin_sin(start));
twin_path_draw_polar(path, start);

for (twin_angle_t a = first; a != last; a += inc)
twin_path_draw(path, twin_cos(a), twin_sin(a));
twin_path_draw_polar(path, a);

if (last != start + extent)
twin_path_draw(path, twin_cos(start + extent),
twin_sin(start + extent));
twin_path_draw_polar(path, start + extent);

twin_path_set_matrix(path, save);
}
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216 changes: 69 additions & 147 deletions src/trig.c
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Original file line number Diff line number Diff line change
Expand Up @@ -6,167 +6,28 @@

#include "twin_private.h"

#define TWIN_LOG2_SIN 10

/*
* construction:
int s = 10; for (int i = 0; i < (1 << s); i++) {
if (i % 8 == 0) printf ("\n ");
printf (" 0x%04x,", floor (sin(pi/2 * i / (1 << s)) * 2**16 + 0.5));
}
*/

static const uint16_t _twin_sin_table[1 << TWIN_LOG2_SIN] = {
0x0000, 0x0065, 0x00c9, 0x012e, 0x0192, 0x01f7, 0x025b, 0x02c0, 0x0324,
0x0389, 0x03ed, 0x0452, 0x04b6, 0x051b, 0x057f, 0x05e4, 0x0648, 0x06ad,
0x0711, 0x0776, 0x07da, 0x083f, 0x08a3, 0x0908, 0x096c, 0x09d1, 0x0a35,
0x0a9a, 0x0afe, 0x0b62, 0x0bc7, 0x0c2b, 0x0c90, 0x0cf4, 0x0d59, 0x0dbd,
0x0e21, 0x0e86, 0x0eea, 0x0f4e, 0x0fb3, 0x1017, 0x107b, 0x10e0, 0x1144,
0x11a8, 0x120d, 0x1271, 0x12d5, 0x1339, 0x139e, 0x1402, 0x1466, 0x14ca,
0x152e, 0x1593, 0x15f7, 0x165b, 0x16bf, 0x1723, 0x1787, 0x17eb, 0x1850,
0x18b4, 0x1918, 0x197c, 0x19e0, 0x1a44, 0x1aa8, 0x1b0c, 0x1b70, 0x1bd4,
0x1c38, 0x1c9b, 0x1cff, 0x1d63, 0x1dc7, 0x1e2b, 0x1e8f, 0x1ef3, 0x1f56,
0x1fba, 0x201e, 0x2082, 0x20e5, 0x2149, 0x21ad, 0x2210, 0x2274, 0x22d7,
0x233b, 0x239f, 0x2402, 0x2466, 0x24c9, 0x252d, 0x2590, 0x25f4, 0x2657,
0x26ba, 0x271e, 0x2781, 0x27e4, 0x2848, 0x28ab, 0x290e, 0x2971, 0x29d5,
0x2a38, 0x2a9b, 0x2afe, 0x2b61, 0x2bc4, 0x2c27, 0x2c8a, 0x2ced, 0x2d50,
0x2db3, 0x2e16, 0x2e79, 0x2edc, 0x2f3f, 0x2fa1, 0x3004, 0x3067, 0x30ca,
0x312c, 0x318f, 0x31f1, 0x3254, 0x32b7, 0x3319, 0x337c, 0x33de, 0x3440,
0x34a3, 0x3505, 0x3568, 0x35ca, 0x362c, 0x368e, 0x36f1, 0x3753, 0x37b5,
0x3817, 0x3879, 0x38db, 0x393d, 0x399f, 0x3a01, 0x3a63, 0x3ac5, 0x3b27,
0x3b88, 0x3bea, 0x3c4c, 0x3cae, 0x3d0f, 0x3d71, 0x3dd2, 0x3e34, 0x3e95,
0x3ef7, 0x3f58, 0x3fba, 0x401b, 0x407c, 0x40de, 0x413f, 0x41a0, 0x4201,
0x4262, 0x42c3, 0x4324, 0x4385, 0x43e6, 0x4447, 0x44a8, 0x4509, 0x456a,
0x45cb, 0x462b, 0x468c, 0x46ec, 0x474d, 0x47ae, 0x480e, 0x486f, 0x48cf,
0x492f, 0x4990, 0x49f0, 0x4a50, 0x4ab0, 0x4b10, 0x4b71, 0x4bd1, 0x4c31,
0x4c90, 0x4cf0, 0x4d50, 0x4db0, 0x4e10, 0x4e70, 0x4ecf, 0x4f2f, 0x4f8e,
0x4fee, 0x504d, 0x50ad, 0x510c, 0x516c, 0x51cb, 0x522a, 0x5289, 0x52e8,
0x5348, 0x53a7, 0x5406, 0x5464, 0x54c3, 0x5522, 0x5581, 0x55e0, 0x563e,
0x569d, 0x56fc, 0x575a, 0x57b9, 0x5817, 0x5875, 0x58d4, 0x5932, 0x5990,
0x59ee, 0x5a4c, 0x5aaa, 0x5b08, 0x5b66, 0x5bc4, 0x5c22, 0x5c80, 0x5cde,
0x5d3b, 0x5d99, 0x5df6, 0x5e54, 0x5eb1, 0x5f0f, 0x5f6c, 0x5fc9, 0x6026,
0x6084, 0x60e1, 0x613e, 0x619b, 0x61f8, 0x6254, 0x62b1, 0x630e, 0x636b,
0x63c7, 0x6424, 0x6480, 0x64dd, 0x6539, 0x6595, 0x65f2, 0x664e, 0x66aa,
0x6706, 0x6762, 0x67be, 0x681a, 0x6876, 0x68d1, 0x692d, 0x6989, 0x69e4,
0x6a40, 0x6a9b, 0x6af6, 0x6b52, 0x6bad, 0x6c08, 0x6c63, 0x6cbe, 0x6d19,
0x6d74, 0x6dcf, 0x6e2a, 0x6e85, 0x6edf, 0x6f3a, 0x6f94, 0x6fef, 0x7049,
0x70a3, 0x70fe, 0x7158, 0x71b2, 0x720c, 0x7266, 0x72c0, 0x731a, 0x7373,
0x73cd, 0x7427, 0x7480, 0x74da, 0x7533, 0x758d, 0x75e6, 0x763f, 0x7698,
0x76f1, 0x774a, 0x77a3, 0x77fc, 0x7855, 0x78ad, 0x7906, 0x795f, 0x79b7,
0x7a10, 0x7a68, 0x7ac0, 0x7b18, 0x7b70, 0x7bc8, 0x7c20, 0x7c78, 0x7cd0,
0x7d28, 0x7d7f, 0x7dd7, 0x7e2f, 0x7e86, 0x7edd, 0x7f35, 0x7f8c, 0x7fe3,
0x803a, 0x8091, 0x80e8, 0x813f, 0x8195, 0x81ec, 0x8243, 0x8299, 0x82f0,
0x8346, 0x839c, 0x83f2, 0x8449, 0x849f, 0x84f5, 0x854a, 0x85a0, 0x85f6,
0x864c, 0x86a1, 0x86f7, 0x874c, 0x87a1, 0x87f6, 0x884c, 0x88a1, 0x88f6,
0x894a, 0x899f, 0x89f4, 0x8a49, 0x8a9d, 0x8af2, 0x8b46, 0x8b9a, 0x8bef,
0x8c43, 0x8c97, 0x8ceb, 0x8d3f, 0x8d93, 0x8de6, 0x8e3a, 0x8e8d, 0x8ee1,
0x8f34, 0x8f88, 0x8fdb, 0x902e, 0x9081, 0x90d4, 0x9127, 0x9179, 0x91cc,
0x921f, 0x9271, 0x92c4, 0x9316, 0x9368, 0x93ba, 0x940c, 0x945e, 0x94b0,
0x9502, 0x9554, 0x95a5, 0x95f7, 0x9648, 0x969a, 0x96eb, 0x973c, 0x978d,
0x97de, 0x982f, 0x9880, 0x98d0, 0x9921, 0x9972, 0x99c2, 0x9a12, 0x9a63,
0x9ab3, 0x9b03, 0x9b53, 0x9ba3, 0x9bf2, 0x9c42, 0x9c92, 0x9ce1, 0x9d31,
0x9d80, 0x9dcf, 0x9e1e, 0x9e6d, 0x9ebc, 0x9f0b, 0x9f5a, 0x9fa8, 0x9ff7,
0xa045, 0xa094, 0xa0e2, 0xa130, 0xa17e, 0xa1cc, 0xa21a, 0xa268, 0xa2b5,
0xa303, 0xa350, 0xa39e, 0xa3eb, 0xa438, 0xa485, 0xa4d2, 0xa51f, 0xa56c,
0xa5b8, 0xa605, 0xa652, 0xa69e, 0xa6ea, 0xa736, 0xa782, 0xa7ce, 0xa81a,
0xa866, 0xa8b2, 0xa8fd, 0xa949, 0xa994, 0xa9df, 0xaa2a, 0xaa76, 0xaac1,
0xab0b, 0xab56, 0xaba1, 0xabeb, 0xac36, 0xac80, 0xacca, 0xad14, 0xad5e,
0xada8, 0xadf2, 0xae3c, 0xae85, 0xaecf, 0xaf18, 0xaf62, 0xafab, 0xaff4,
0xb03d, 0xb086, 0xb0ce, 0xb117, 0xb160, 0xb1a8, 0xb1f0, 0xb239, 0xb281,
0xb2c9, 0xb311, 0xb358, 0xb3a0, 0xb3e8, 0xb42f, 0xb477, 0xb4be, 0xb505,
0xb54c, 0xb593, 0xb5da, 0xb620, 0xb667, 0xb6ad, 0xb6f4, 0xb73a, 0xb780,
0xb7c6, 0xb80c, 0xb852, 0xb898, 0xb8dd, 0xb923, 0xb968, 0xb9ae, 0xb9f3,
0xba38, 0xba7d, 0xbac1, 0xbb06, 0xbb4b, 0xbb8f, 0xbbd4, 0xbc18, 0xbc5c,
0xbca0, 0xbce4, 0xbd28, 0xbd6b, 0xbdaf, 0xbdf2, 0xbe36, 0xbe79, 0xbebc,
0xbeff, 0xbf42, 0xbf85, 0xbfc7, 0xc00a, 0xc04c, 0xc08f, 0xc0d1, 0xc113,
0xc155, 0xc197, 0xc1d8, 0xc21a, 0xc25c, 0xc29d, 0xc2de, 0xc31f, 0xc360,
0xc3a1, 0xc3e2, 0xc423, 0xc463, 0xc4a4, 0xc4e4, 0xc524, 0xc564, 0xc5a4,
0xc5e4, 0xc624, 0xc663, 0xc6a3, 0xc6e2, 0xc721, 0xc761, 0xc7a0, 0xc7de,
0xc81d, 0xc85c, 0xc89a, 0xc8d9, 0xc917, 0xc955, 0xc993, 0xc9d1, 0xca0f,
0xca4d, 0xca8a, 0xcac7, 0xcb05, 0xcb42, 0xcb7f, 0xcbbc, 0xcbf9, 0xcc35,
0xcc72, 0xccae, 0xcceb, 0xcd27, 0xcd63, 0xcd9f, 0xcddb, 0xce17, 0xce52,
0xce8e, 0xcec9, 0xcf04, 0xcf3f, 0xcf7a, 0xcfb5, 0xcff0, 0xd02a, 0xd065,
0xd09f, 0xd0d9, 0xd113, 0xd14d, 0xd187, 0xd1c1, 0xd1fa, 0xd234, 0xd26d,
0xd2a6, 0xd2df, 0xd318, 0xd351, 0xd38a, 0xd3c2, 0xd3fb, 0xd433, 0xd46b,
0xd4a3, 0xd4db, 0xd513, 0xd54b, 0xd582, 0xd5ba, 0xd5f1, 0xd628, 0xd65f,
0xd696, 0xd6cd, 0xd703, 0xd73a, 0xd770, 0xd7a6, 0xd7dc, 0xd812, 0xd848,
0xd87e, 0xd8b4, 0xd8e9, 0xd91e, 0xd954, 0xd989, 0xd9be, 0xd9f2, 0xda27,
0xda5c, 0xda90, 0xdac4, 0xdaf8, 0xdb2c, 0xdb60, 0xdb94, 0xdbc8, 0xdbfb,
0xdc2f, 0xdc62, 0xdc95, 0xdcc8, 0xdcfb, 0xdd2d, 0xdd60, 0xdd92, 0xddc5,
0xddf7, 0xde29, 0xde5b, 0xde8c, 0xdebe, 0xdef0, 0xdf21, 0xdf52, 0xdf83,
0xdfb4, 0xdfe5, 0xe016, 0xe046, 0xe077, 0xe0a7, 0xe0d7, 0xe107, 0xe137,
0xe167, 0xe196, 0xe1c6, 0xe1f5, 0xe224, 0xe253, 0xe282, 0xe2b1, 0xe2df,
0xe30e, 0xe33c, 0xe36b, 0xe399, 0xe3c7, 0xe3f4, 0xe422, 0xe450, 0xe47d,
0xe4aa, 0xe4d7, 0xe504, 0xe531, 0xe55e, 0xe58b, 0xe5b7, 0xe5e3, 0xe610,
0xe63c, 0xe667, 0xe693, 0xe6bf, 0xe6ea, 0xe716, 0xe741, 0xe76c, 0xe797,
0xe7c2, 0xe7ec, 0xe817, 0xe841, 0xe86b, 0xe895, 0xe8bf, 0xe8e9, 0xe913,
0xe93c, 0xe966, 0xe98f, 0xe9b8, 0xe9e1, 0xea0a, 0xea32, 0xea5b, 0xea83,
0xeaab, 0xead4, 0xeafc, 0xeb23, 0xeb4b, 0xeb73, 0xeb9a, 0xebc1, 0xebe8,
0xec0f, 0xec36, 0xec5d, 0xec83, 0xecaa, 0xecd0, 0xecf6, 0xed1c, 0xed42,
0xed68, 0xed8d, 0xedb3, 0xedd8, 0xedfd, 0xee22, 0xee47, 0xee6b, 0xee90,
0xeeb4, 0xeed9, 0xeefd, 0xef21, 0xef45, 0xef68, 0xef8c, 0xefaf, 0xefd2,
0xeff5, 0xf018, 0xf03b, 0xf05e, 0xf080, 0xf0a3, 0xf0c5, 0xf0e7, 0xf109,
0xf12b, 0xf14c, 0xf16e, 0xf18f, 0xf1b1, 0xf1d2, 0xf1f3, 0xf213, 0xf234,
0xf254, 0xf275, 0xf295, 0xf2b5, 0xf2d5, 0xf2f5, 0xf314, 0xf334, 0xf353,
0xf372, 0xf391, 0xf3b0, 0xf3cf, 0xf3ed, 0xf40c, 0xf42a, 0xf448, 0xf466,
0xf484, 0xf4a2, 0xf4bf, 0xf4dd, 0xf4fa, 0xf517, 0xf534, 0xf551, 0xf56e,
0xf58a, 0xf5a6, 0xf5c3, 0xf5df, 0xf5fb, 0xf616, 0xf632, 0xf64e, 0xf669,
0xf684, 0xf69f, 0xf6ba, 0xf6d5, 0xf6ef, 0xf70a, 0xf724, 0xf73e, 0xf758,
0xf772, 0xf78c, 0xf7a5, 0xf7bf, 0xf7d8, 0xf7f1, 0xf80a, 0xf823, 0xf83b,
0xf854, 0xf86c, 0xf885, 0xf89d, 0xf8b4, 0xf8cc, 0xf8e4, 0xf8fb, 0xf913,
0xf92a, 0xf941, 0xf958, 0xf96e, 0xf985, 0xf99b, 0xf9b2, 0xf9c8, 0xf9de,
0xf9f3, 0xfa09, 0xfa1f, 0xfa34, 0xfa49, 0xfa5e, 0xfa73, 0xfa88, 0xfa9c,
0xfab1, 0xfac5, 0xfad9, 0xfaed, 0xfb01, 0xfb15, 0xfb28, 0xfb3c, 0xfb4f,
0xfb62, 0xfb75, 0xfb88, 0xfb9a, 0xfbad, 0xfbbf, 0xfbd1, 0xfbe3, 0xfbf5,
0xfc07, 0xfc18, 0xfc2a, 0xfc3b, 0xfc4c, 0xfc5d, 0xfc6e, 0xfc7f, 0xfc8f,
0xfca0, 0xfcb0, 0xfcc0, 0xfcd0, 0xfcdf, 0xfcef, 0xfcfe, 0xfd0e, 0xfd1d,
0xfd2c, 0xfd3b, 0xfd49, 0xfd58, 0xfd66, 0xfd74, 0xfd83, 0xfd90, 0xfd9e,
0xfdac, 0xfdb9, 0xfdc7, 0xfdd4, 0xfde1, 0xfdee, 0xfdfa, 0xfe07, 0xfe13,
0xfe1f, 0xfe2b, 0xfe37, 0xfe43, 0xfe4f, 0xfe5a, 0xfe66, 0xfe71, 0xfe7c,
0xfe87, 0xfe91, 0xfe9c, 0xfea6, 0xfeb0, 0xfeba, 0xfec4, 0xfece, 0xfed8,
0xfee1, 0xfeeb, 0xfef4, 0xfefd, 0xff06, 0xff0e, 0xff17, 0xff1f, 0xff28,
0xff30, 0xff38, 0xff3f, 0xff47, 0xff4e, 0xff56, 0xff5d, 0xff64, 0xff6b,
0xff71, 0xff78, 0xff7e, 0xff85, 0xff8b, 0xff91, 0xff96, 0xff9c, 0xffa2,
0xffa7, 0xffac, 0xffb1, 0xffb6, 0xffbb, 0xffbf, 0xffc4, 0xffc8, 0xffcc,
0xffd0, 0xffd4, 0xffd7, 0xffdb, 0xffde, 0xffe1, 0xffe4, 0xffe7, 0xffea,
0xffec, 0xffef, 0xfff1, 0xfff3, 0xfff5, 0xfff7, 0xfff8, 0xfffa, 0xfffb,
0xfffc, 0xfffd, 0xfffe, 0xffff, 0xffff, 0xffff, 0xffff,
};

/*
* angles are measured from -2048 .. 2048
*/

twin_fixed_t twin_sin(twin_angle_t a)
{
twin_fixed_t sin;

/* limit to [0..360) */
a = a & (TWIN_ANGLE_360 - 1);
/* special case for 90 degrees - no room in table */
if ((a & ~(TWIN_ANGLE_180)) == TWIN_ANGLE_90)
sin = TWIN_FIXED_ONE;
else {
/* mirror second and third quadrant values across y axis */
if (a & TWIN_ANGLE_90)
a = TWIN_ANGLE_180 - a;
sin = _twin_sin_table[a & (TWIN_ANGLE_90 - 1)];
}
/* mirror third and fourth quadrant values across x axis */
if (a & TWIN_ANGLE_180)
sin = -sin;
return sin;
twin_fixed_t sin_val = 0;
twin_sincos(a, &sin_val, NULL);
return sin_val;
}

twin_fixed_t twin_cos(twin_angle_t a)
{
return twin_sin(a + TWIN_ANGLE_90);
twin_fixed_t cos_val = 0;
twin_sincos(a, NULL, &cos_val);
return cos_val;
}

twin_fixed_t twin_tan(twin_angle_t a)
{
twin_fixed_t s = twin_sin(a);
twin_fixed_t c = twin_cos(a);
twin_fixed_t s, c;
twin_sincos(a, &s, &c);

if (c == 0) {
if (s > 0)
Expand All @@ -178,3 +39,64 @@ twin_fixed_t twin_tan(twin_angle_t a)
return 0;
return ((s << 15) / c) << 1;
}

static inline twin_fixed_t sin_poly(twin_angle_t x)
{
/* S(x) = x * 2^(-n) * (A1 - 2 ^ (q-p) * x * (2^-n) * x * 2^(-n) * (B1 - 2 ^
* (-r) * x * 2 ^ (-n) * C1 * x)) * 2 ^ (a-q)
* @n: the angle scale
* @A: the amplitude
* @p,q,r: the scaling factor
*
* A1 = 2^q * a5, B1 = 2 ^ p * b5, C1 = 2 ^ (r+p-n) * c5
* where a5, b5, c5 are the coefficients for 5th-order polynomial
* a5 = 4 * (3 / pi - 9 / 16)
* b5 = 2 * a5 - 5 / 2
* c5 = a5 - 3 / 2
*/
const uint64_t A = 16, n = 10, p = 32, q = 31, r = 3;
const uint64_t A1 = 3370945099, B1 = 2746362156, C1 = 2339369;
uint64_t y = (C1 * x) >> n;
y = B1 - ((x * y) >> r);
y = x * (y >> n);
y = x * (y >> n);
y = A1 - (y >> (p - q));
y = x * (y >> n);
y = (y + (1UL << (q - A - 1))) >> (q - A); // Rounding
return y;
}

void twin_sincos(twin_angle_t a, twin_fixed_t *sin, twin_fixed_t *cos)
{
twin_fixed_t sin_val = 0, cos_val = 0;

/* limit to [0..360) */
a = a & (TWIN_ANGLE_360 - 1);
int c = a > TWIN_ANGLE_90 && a < TWIN_ANGLE_270;
/* special case for 90 degrees */
if ((a & ~(TWIN_ANGLE_180)) == TWIN_ANGLE_90) {
sin_val = TWIN_FIXED_ONE;
cos_val = 0;
} else {
/* mirror second and third quadrant values across y axis */
if (a & TWIN_ANGLE_90)
a = TWIN_ANGLE_180 - a;
twin_angle_t x = a & (TWIN_ANGLE_90 - 1);
if (sin)
sin_val = sin_poly(x);
if (cos)
cos_val = sin_poly(TWIN_ANGLE_90 - x);
}
if (sin) {
/* mirror third and fourth quadrant values across x axis */
if (a & TWIN_ANGLE_180)
sin_val = -sin_val;
*sin = sin_val;
}
if (cos) {
/* mirror first and fourth quadrant values across y axis */
if (c)
cos_val = -cos_val;
*cos = cos_val;
}
}

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