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#ifndef __INC_LIB8TION_TRIG_H
#define __INC_LIB8TION_TRIG_H
///@ingroup lib8tion
///@defgroup Trig Fast trig functions
/// Fast 8 and 16-bit approximations of sin(x) and cos(x).
/// Don't use these approximations for calculating the
/// trajectory of a rocket to Mars, but they're great
/// for art projects and LED displays.
///
/// On Arduino/AVR, the 16-bit approximation is more than
/// 10X faster than floating point sin(x) and cos(x), while
/// the 8-bit approximation is more than 20X faster.
///@{
#if defined(__AVR__)
#define sin16 sin16_avr
#else
#define sin16 sin16_C
#endif
/// Fast 16-bit approximation of sin(x). This approximation never varies more than
/// 0.69% from the floating point value you'd get by doing
///
/// float s = sin(x) * 32767.0;
///
/// @param theta input angle from 0-65535
/// @returns sin of theta, value between -32767 to 32767.
LIB8STATIC int16_t sin16_avr( uint16_t theta )
{
static const uint8_t data[] =
{ 0, 0, 49, 0, 6393%256, 6393/256, 48, 0,
12539%256, 12539/256, 44, 0, 18204%256, 18204/256, 38, 0,
23170%256, 23170/256, 31, 0, 27245%256, 27245/256, 23, 0,
30273%256, 30273/256, 14, 0, 32137%256, 32137/256, 4 /*,0*/ };
uint16_t offset = (theta & 0x3FFF);
// AVR doesn't have a multi-bit shift instruction,
// so if we say "offset >>= 3", gcc makes a tiny loop.
// Inserting empty volatile statements between each
// bit shift forces gcc to unroll the loop.
offset >>= 1; // 0..8191
asm volatile("");
offset >>= 1; // 0..4095
asm volatile("");
offset >>= 1; // 0..2047
if( theta & 0x4000 ) offset = 2047 - offset;
uint8_t sectionX4;
sectionX4 = offset / 256;
sectionX4 *= 4;
uint8_t m;
union {
uint16_t b;
struct {
uint8_t blo;
uint8_t bhi;
};
} u;
//in effect u.b = blo + (256 * bhi);
u.blo = data[ sectionX4 ];
u.bhi = data[ sectionX4 + 1];
m = data[ sectionX4 + 2];
uint8_t secoffset8 = (uint8_t)(offset) / 2;
uint16_t mx = m * secoffset8;
int16_t y = mx + u.b;
if( theta & 0x8000 ) y = -y;
return y;
}
/// Fast 16-bit approximation of sin(x). This approximation never varies more than
/// 0.69% from the floating point value you'd get by doing
///
/// float s = sin(x) * 32767.0;
///
/// @param theta input angle from 0-65535
/// @returns sin of theta, value between -32767 to 32767.
LIB8STATIC int16_t sin16_C( uint16_t theta )
{
static const uint16_t base[] =
{ 0, 6393, 12539, 18204, 23170, 27245, 30273, 32137 };
static const uint8_t slope[] =
{ 49, 48, 44, 38, 31, 23, 14, 4 };
uint16_t offset = (theta & 0x3FFF) >> 3; // 0..2047
if( theta & 0x4000 ) offset = 2047 - offset;
uint8_t section = offset / 256; // 0..7
uint16_t b = base[section];
uint8_t m = slope[section];
uint8_t secoffset8 = (uint8_t)(offset) / 2;
uint16_t mx = m * secoffset8;
int16_t y = mx + b;
if( theta & 0x8000 ) y = -y;
return y;
}
/// Fast 16-bit approximation of cos(x). This approximation never varies more than
/// 0.69% from the floating point value you'd get by doing
///
/// float s = cos(x) * 32767.0;
///
/// @param theta input angle from 0-65535
/// @returns sin of theta, value between -32767 to 32767.
LIB8STATIC int16_t cos16( uint16_t theta)
{
return sin16( theta + 16384);
}
///////////////////////////////////////////////////////////////////////
// sin8 & cos8
// Fast 8-bit approximations of sin(x) & cos(x).
// Input angle is an unsigned int from 0-255.
// Output is an unsigned int from 0 to 255.
//
// This approximation can vary to to 2%
// from the floating point value you'd get by doing
// float s = (sin( x ) * 128.0) + 128;
//
// Don't use this approximation for calculating the
// "real" trigonometric calculations, but it's great
// for art projects and LED displays.
//
// On Arduino/AVR, this approximation is more than
// 20X faster than floating point sin(x) and cos(x)
#if defined(__AVR__) && !defined(LIB8_ATTINY)
#define sin8 sin8_avr
#else
#define sin8 sin8_C
#endif
const uint8_t b_m16_interleave[] = { 0, 49, 49, 41, 90, 27, 117, 10 };
/// Fast 8-bit approximation of sin(x). This approximation never varies more than
/// 2% from the floating point value you'd get by doing
///
/// float s = (sin(x) * 128.0) + 128;
///
/// @param theta input angle from 0-255
/// @returns sin of theta, value between 0 and 255
LIB8STATIC uint8_t sin8_avr( uint8_t theta)
{
uint8_t offset = theta;
asm volatile(
"sbrc %[theta],6 \n\t"
"com %[offset] \n\t"
: [theta] "+r" (theta), [offset] "+r" (offset)
);
offset &= 0x3F; // 0..63
uint8_t secoffset = offset & 0x0F; // 0..15
if( theta & 0x40) secoffset++;
uint8_t m16; uint8_t b;
uint8_t section = offset >> 4; // 0..3
uint8_t s2 = section * 2;
const uint8_t* p = b_m16_interleave;
p += s2;
b = *p;
p++;
m16 = *p;
uint8_t mx;
uint8_t xr1;
asm volatile(
"mul %[m16],%[secoffset] \n\t"
"mov %[mx],r0 \n\t"
"mov %[xr1],r1 \n\t"
"eor r1, r1 \n\t"
"swap %[mx] \n\t"
"andi %[mx],0x0F \n\t"
"swap %[xr1] \n\t"
"andi %[xr1], 0xF0 \n\t"
"or %[mx], %[xr1] \n\t"
: [mx] "=d" (mx), [xr1] "=d" (xr1)
: [m16] "d" (m16), [secoffset] "d" (secoffset)
);
int8_t y = mx + b;
if( theta & 0x80 ) y = -y;
y += 128;
return y;
}
/// Fast 8-bit approximation of sin(x). This approximation never varies more than
/// 2% from the floating point value you'd get by doing
///
/// float s = (sin(x) * 128.0) + 128;
///
/// @param theta input angle from 0-255
/// @returns sin of theta, value between 0 and 255
LIB8STATIC uint8_t sin8_C( uint8_t theta)
{
uint8_t offset = theta;
if( theta & 0x40 ) {
offset = (uint8_t)255 - offset;
}
offset &= 0x3F; // 0..63
uint8_t secoffset = offset & 0x0F; // 0..15
if( theta & 0x40) secoffset++;
uint8_t section = offset >> 4; // 0..3
uint8_t s2 = section * 2;
const uint8_t* p = b_m16_interleave;
p += s2;
uint8_t b = *p;
p++;
uint8_t m16 = *p;
uint8_t mx = (m16 * secoffset) >> 4;
int8_t y = mx + b;
if( theta & 0x80 ) y = -y;
y += 128;
return y;
}
/// Fast 8-bit approximation of cos(x). This approximation never varies more than
/// 2% from the floating point value you'd get by doing
///
/// float s = (cos(x) * 128.0) + 128;
///
/// @param theta input angle from 0-255
/// @returns sin of theta, value between 0 and 255
LIB8STATIC uint8_t cos8( uint8_t theta)
{
return sin8( theta + 64);
}
///@}
#endif
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