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-rw-r--r--fixpow.c169
1 files changed, 93 insertions, 76 deletions
diff --git a/fixpow.c b/fixpow.c
index d81881f..cb687b4 100644
--- a/fixpow.c
+++ b/fixpow.c
@@ -11,23 +11,27 @@
11#include <stdlib.h> 11#include <stdlib.h>
12#include <math.h> 12#include <math.h>
13 13
14#define TO_Q16(X) ((int32_t)(65536.f*(X))) 14#define TO_Q16(X) ((int32_t)(65536.*(X)))
15#define FROM_Q16(X) ((double)((X)/65536.f)) 15#define TO_Q26(X) ((int32_t)(67108864.*(X)))
16#define FROM_Q16(X) ((double)((X)/65536.))
17#define FROM_Q26(X) ((double)((x)/67108864.))
16 18
17// It works as follows: 19/*
18// P = pow_QX_QY_QP( x, y ) == exp( ln(x) * y ) 20 It works as follows:
21 P = pow_QX_QY_QP( x, y ) == exp( ln(x) * y )
19 22
20// Where if x is negative: 23 Where if x is negative:
21// * y must have no fractional part, 24 * y must have no fractional part,
22// * the result's sign is the lowest integral bit of y 25 * the result's sign is the lowest integral bit of y
23 26
24// We note that in every standard Q representation ln(x) will not exceed 27 We note that in every standard Q representation ln(x) will not exceed
25// the value 22, so that we can safely work with a Q26 representation. 28 the value 22, so that we can safely work with a Q26 representation.
26// 29
27// if ln(x) * y would overflow in that representation, so would 30 if ln(x) * y would overflow in that representation, so would
28// exp( ln(x) * y ) in Q16. 31 exp( ln(x) * y ) in Q16.
29// 32
30// Finally exp() is calculated in the domain specified by script 33 Finally exp() is calculated in the domain specified by script
34*/
31 35
32#if 0 36#if 0
33 The table[tm], generated from different versions of 37 The table[tm], generated from different versions of
@@ -72,19 +76,19 @@
72 76
73for negative values: 77for negative values:
74 78
75-00,374693449441410697531 017FAFA3 x-= ( x >> 2 ) + ( x >> 4 ) 79-00,374693449441410697531 x-= ( x >> 2 ) + ( x >> 4 )
76-00,207639364778244489562 00D49F69 x-= ( x >> 2 ) - ( x >> 4 ) 80-00,207639364778244489562 x-= ( x >> 2 ) - ( x >> 4 )
77-00,115831815525121700761 00769C9D x-= ( x >> 3 ) - ( x >> 6 ) 81-00,115831815525121700761 x-= ( x >> 3 ) - ( x >> 6 )
78-00,060380510988907482028 003DD463 x-= ( x >> 4 ) - ( x >> 8 ) 82-00,060380510988907482028 x-= ( x >> 4 ) - ( x >> 8 )
79-00,031748698314580298119 002082BB x-= ( x >> 5 ) 83-00,031748698314580298119 x-= ( x >> 5 )
80-00,015996403602177928366 0010615C x-= ( x >> 6 ) + ( x >> 12 ) 84-00,015996403602177928366 x-= ( x >> 6 ) + ( x >> 12 )
81-00,008089270731616965762 0008488D x-= ( x >> 7 ) + ( x >> 12 ) 85-00,008089270731616965762 x-= ( x >> 7 ) + ( x >> 12 )
82-00,004159027401785260480 00044243 x-= ( x >> 8 ) + ( x >> 12 ) 86-00,004159027401785260480 x-= ( x >> 8 ) + ( x >> 12 )
83-00,003423823349553609900 00038188 x-= ( x >> 8 ) - ( x >> 11 ) 87-00,003423823349553609900 x-= ( x >> 8 ) - ( x >> 11 )
84-00,001832733117311761669 0001E070 x-= ( x >> 9 ) - ( x >> 13 ) 88-00,001832733117311761669 x-= ( x >> 9 ) - ( x >> 13 )
85-00,000961766059678620302 0000FC1F x-= ( x >> 10 ) - ( x >> 16 ) 89-00,000961766059678620302 x-= ( x >> 10 ) - ( x >> 16 )
86-00,000484583944571210926 00007F07 x-= ( x >> 11 ) - ( x >> 18 ) 90-00,000484583944571210926 x-= ( x >> 11 ) - ( x >> 18 )
87-00,000243216525424976433 00003FC1 x-= ( x >> 12 ) - ( x >> 20 ) 91-00,000243216525424976433 x-= ( x >> 12 ) - ( x >> 20 )
88 92
89#endif 93#endif
90 94
@@ -92,53 +96,60 @@ for negative values:
92static int32_t fixlog( int32_t x ) { 96static int32_t fixlog( int32_t x ) {
93 int32_t t,y; 97 int32_t t,y;
94 98
95 y = 0x2996BD9E; // ln(2^15) << 26 99 // Assume an x in Q21 (shift by 15) and thus start with y with log(2^15)
96 if(x<0x00008000) x<<=16, y-= 0x2C5C85FD; 100 y = TO_Q26(10.39720770839918);
97 if(x<0x00800000) x<<= 8, y-= 0x162E42FE; 101 if(x<0x00008000) x<<=16, y-= TO_Q26(11.09035488895912); /* log(65536) */
98 if(x<0x08000000) x<<= 4, y-= 0x0B17217F; 102 if(x<0x00800000) x<<= 8, y-= TO_Q26(5.545177444479562); /* log(256) */
99 if(x<0x20000000) x<<= 2, y-= 0x058B90BF; 103 if(x<0x08000000) x<<= 4, y-= TO_Q26(2.772588722239781); /* log(16) */
100 if(x<0x40000000) x<<= 1, y-= 0x02C5C85F; 104 if(x<0x20000000) x<<= 2, y-= TO_Q26(1.386294361119891); /* log(4) */
101 t=x+(x>>1); if((t&0x80000000)==0) x=t, y-= 0x019F323E; 105 if(x<0x40000000) x<<= 1, y-= TO_Q26(0.693147180559945); /* log(2) */
102 t=x+(x>>2); if((t&0x80000000)==0) x=t, y-= 0x00E47FBE; 106 t=x+(x>>1); if((t&0x80000000)==0) x=t, y-= TO_Q26(0.405465108108164); /* log(1+1/2) */
103 t=x+(x>>3); if((t&0x80000000)==0) x=t, y-= 0x00789C1D; 107 t=x+(x>>2); if((t&0x80000000)==0) x=t, y-= TO_Q26(0.22314355131421); /* log(1+1/4) */
104 t=x+(x>>4); if((t&0x80000000)==0) x=t, y-= 0x003E1461; 108 t=x+(x>>3); if((t&0x80000000)==0) x=t, y-= TO_Q26(0.117783035656384); /* log(1+1/8) */
105 t=x+(x>>5); if((t&0x80000000)==0) x=t, y-= 0x001F829B; 109 t=x+(x>>4); if((t&0x80000000)==0) x=t, y-= TO_Q26(0.060624621816435); /* log(1+1/16) */
106 t=x+(x>>6); if((t&0x80000000)==0) x=t, y-= 0x000FE054; 110 t=x+(x>>5); if((t&0x80000000)==0) x=t, y-= TO_Q26(0.030771658666754); /* log(1+1/32) */
107 t=x+(x>>7); if((t&0x80000000)==0) x=t, y-= 0x0007F80A; 111 t=x+(x>>6); if((t&0x80000000)==0) x=t, y-= TO_Q26(0.015504186535965); /* log(1+1/64) */
108 t=x+(x>>8); if((t&0x80000000)==0) x=t, y-= 0x0003FE01; 112 t=x+(x>>7); if((t&0x80000000)==0) x=t, y-= TO_Q26(0.007782140442055); /* log(1+1/128) */
109 t=x+(x>>9); if((t&0x80000000)==0) x=t, y-= 0x0001FF80; 113 t=x+(x>>8); if((t&0x80000000)==0) x=t, y-= TO_Q26(0.003898640415657); /* log(1+1/256) */
110 t=x+(x>>10);if((t&0x80000000)==0) x=t, y-= 0x0000FFE0; 114 t=x+(x>>9); if((t&0x80000000)==0) x=t, y-= TO_Q26(0.001951220131262); /* log(1+1/512) */
111 t=x+(x>>11);if((t&0x80000000)==0) x=t, y-= 0x00007FF8; 115 t=x+(x>>10);if((t&0x80000000)==0) x=t, y-= TO_Q26(0.000976085973055); /* log(1+1/1024) */
112 t=x+(x>>12);if((t&0x80000000)==0) x=t, y-= 0x00003FFE; 116 t=x+(x>>11);if((t&0x80000000)==0) x=t, y-= TO_Q26(0.000488162079501); /* log(1+1/2048) */
113 t=x+(x>>13);if((t&0x80000000)==0) x=t, y-= 0x00001FFF; 117 t=x+(x>>12);if((t&0x80000000)==0) x=t, y-= TO_Q26(0.000244110827527); /* log(1+1/4096) */
118 t=x+(x>>13);if((t&0x80000000)==0) x=t, y-= TO_Q26(0.000122062862526); /* log(1+1/8192) */
119
120 /* Estimate residual error, log(1-x) which for small x is approx -x */
114 x = 0x80000000-x; 121 x = 0x80000000-x;
115 y-= x>>5; 122
116 return y; 123 /* x has been promoted to Q31, substract residual error in Q26 by shifting down by 5 */
124 return y - ( x >> 5 );
117} 125}
118 126
119// input is Q26, output Q16 127// input is Q26, output Q16
120static uint32_t fixexp(int32_t x) { 128static uint32_t fixexp(int32_t x) {
121 int32_t t; 129 int32_t t;
130 /* Start y with 1.0 */
122 uint32_t y = 0x10000; 131 uint32_t y = 0x10000;
123 132
124 if( (x & 0x80000000) == 0 ) { 133 if( x >= 0 ) {
125 t=x-0x162E42FE; if(t>=0) x=t,y<<=8; 134 t=x-TO_Q26(5.545177444479562); if(t>=0) x=t,y<<=8; /* log(256) */
126 t=x-0x0B17217F; if(t>=0) x=t,y<<=4; 135 t=x-TO_Q26(2.772588722239781); if(t>=0) x=t,y<<=4; /* log(16) */
127 t=x-0x058B90BF; if(t>=0) x=t,y<<=2; 136 t=x-TO_Q26(1.386294361119891); if(t>=0) x=t,y<<=2; /* log(4) */
128 t=x-0x02C5C85F; if(t>=0) x=t,y<<=1; 137 t=x-TO_Q26(0.693147180559945); if(t>=0) x=t,y<<=1; /* log(2) */
129 t=x-0x019F323E; if(t>=0) x=t,y+=y>>1; 138 t=x-TO_Q26(0.405465108108164); if(t>=0) x=t,y+=y>>1; /* log(1+1/2) */
130 t=x-0x00E47FBE; if(t>=0) x=t,y+=y>>2; 139 t=x-TO_Q26(0.22314355131421); if(t>=0) x=t,y+=y>>2; /* log(1+1/4) */
131 t=x-0x00789C1D; if(t>=0) x=t,y+=y>>3; 140 t=x-TO_Q26(0.117783035656384); if(t>=0) x=t,y+=y>>3; /* log(1+1/8) */
132 t=x-0x003E1461; if(t>=0) x=t,y+=y>>4; 141 t=x-TO_Q26(0.060624621816435); if(t>=0) x=t,y+=y>>4; /* log(1+1/16) */
133 t=x-0x001F829B; if(t>=0) x=t,y+=y>>5; 142 t=x-TO_Q26(0.030771658666754); if(t>=0) x=t,y+=y>>5; /* log(1+1/32) */
134 t=x-0x000FE054; if(t>=0) x=t,y+=y>>6; 143 t=x-TO_Q26(0.015504186535965); if(t>=0) x=t,y+=y>>6; /* log(1+1/64) */
135 t=x-0x0007F80A; if(t>=0) x=t,y+=y>>7; 144 t=x-TO_Q26(0.007782140442055); if(t>=0) x=t,y+=y>>7; /* log(1+1/128) */
136 t=x-0x0003FE01; if(t>=0) x=t,y+=y>>8; 145 t=x-TO_Q26(0.003898640415657); if(t>=0) x=t,y+=y>>8; /* log(1+1/256) */
137 t=x-0x0001FF80; if(t>=0) x=t,y+=y>>9; 146 t=x-TO_Q26(0.001951220131262); if(t>=0) x=t,y+=y>>9; /* log(1+1/512) */
138 t=x-0x0000FFE0; if(t>=0) x=t,y+=y>>10; 147 t=x-TO_Q26(0.000976085973055); if(t>=0) x=t,y+=y>>10; /* log(1+1/1024) */
139 t=x-0x00007FF8; if(t>=0) x=t,y+=y>>11; 148 t=x-TO_Q26(0.000488162079501); if(t>=0) x=t,y+=y>>11; /* log(1+1/2048) */
140 t=x-0x00003FFE; if(t>=0) x=t,y+=y>>12; 149 t=x-TO_Q26(0.000244110827527); if(t>=0) x=t,y+=y>>12; /* log(1+1/4096) */
141 t=x-0x00001FFF; if(t>=0) x=t,y+=y>>13; 150 t=x-TO_Q26(0.000122062862526); if(t>=0) x=t,y+=y>>13; /* log(1+1/8192) */
151 /* from here the values in Q26 become approx 2^n, so lets check bits
152 and fix the residual error below */
142 if(x&0x0001000) y+=y>>14; 153 if(x&0x0001000) y+=y>>14;
143 if(x&0x0000800) y+=y>>15; 154 if(x&0x0000800) y+=y>>15;
144 if(x&0x0000400) y+=y>>16; 155 if(x&0x0000400) y+=y>>16;
@@ -192,28 +203,34 @@ static uint32_t fixexp(int32_t x) {
192 203
193int fixpow(int32_t *pow, int32_t base, int32_t exponent ) 204int fixpow(int32_t *pow, int32_t base, int32_t exponent )
194{ 205{
195 int neg = 0;
196 int32_t log_base, res; 206 int32_t log_base, res;
197 int64_t log_base_times_exponent; 207 int64_t log_base_times_exponent;
208 int neg = 0;
198 209
199 if( base < 0 ) { 210 if( base < 0 ) {
200 // negative bases only can have integer exponents 211 /* negative bases only can have integer exponents */
201 if( exponent & 0xffff ) return -1; 212 if( exponent & 0xffff ) return -1;
202 // sign of power of a negative base is determined by 213 /* sign of power of a negative base is determined by
203 // wether exp is even 214 wether exp is even */
204 if( exponent & 0x10000 ) neg = 1; 215 if( exponent & 0x10000 ) neg = 1;
205 base = abs(base); 216 base = abs(base);
206 } 217 }
207 218
208 // To calculate pow(base,exp), we do exp( log(base) * exp ) 219 /* To calculate pow(base,exp), we do exp( log(base) * exp )
209 // which is mathematically the same. log_base is Q26 220 which is mathematically the same. log_base is Q26 */
210 log_base = fixlog( base ); 221 log_base = fixlog( base );
211 log_base_times_exponent = ( (int64_t)log_base * (int64_t)exponent ) >> 16; 222 log_base_times_exponent = ( (int64_t)log_base * (int64_t)exponent ) >> 16;
212 223
213 // fixexp overflows for values > 21,48756259689264 which 224#if 0
214 // in Q26 notation is 1442005916 225 /* In Q0, fixexp overflows for values > 21,48756259689264 which
226 in Q26 notation is 1442005916 */
215 if( log_base_times_exponent > 1442005916 ) 227 if( log_base_times_exponent > 1442005916 )
216 return -2; 228 return -2;
229#endif
230
231 /* In Q16, fixexp overflows for values > 10,39720770839918 */
232 if( log_base_times_exponent > TO_Q26(10,39720770839918) )
233 return -2;
217 234
218 res = (int32_t)log_base_times_exponent; 235 res = (int32_t)log_base_times_exponent;
219 res = fixexp( res ); 236 res = fixexp( res );
@@ -225,8 +242,8 @@ int fixpow(int32_t *pow, int32_t base, int32_t exponent )
225 242
226int main() 243int main()
227{ 244{
228 double base = -.5f; 245 double base = -.5;
229 double exponent = 10.f; 246 double exponent = 10.;
230 int32_t result; 247 int32_t result;
231 int error = fixpow( &result, TO_Q16(base), TO_Q16(exponent) ); 248 int error = fixpow( &result, TO_Q16(base), TO_Q16(exponent) );
232 249