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/* Parts of this software were written by Dirk Engling <erdgeist@erdgeist.org>
Those parts are considered beerware. Prost. Skol. Cheers or whatever.
Original idea and the place where I heavily stole code from is
http://www.quinapalus.com/efunc.html
Dr Mark St John OWEN, E-mail: mail -at- quinapalus -dot- com
*/
#include <stdint.h>
#include <stdio.h>
#include <stdlib.h>
#include <math.h>
#define TO_Q16(X) ((int32_t)(65536.*(X)))
#define TO_Q26(X) ((int32_t)(67108864.*(X)))
#define FROM_Q16(X) ((double)((X)/65536.))
#define FROM_Q26(X) ((double)((X)/67108864.))
/*
It works as follows:
P = pow_QX_QY_QP( x, y ) == exp( ln(x) * y )
Where if x is negative:
* y must have no fractional part,
* the result's sign is the lowest integral bit of y
We note that in every standard Q representation ln(x) will not exceed
the value 22, so that we can safely work with a Q26 representation.
if ln(x) * y would overflow in that representation, so would
exp( ln(x) * y ) in Q16.
Finally exp() is calculated in the domain specified by script
*/
#if 0
The table[tm], generated from different versions of
generate_table, though.
+00,000000000465661287199 x = x + ( x >> 31 )
+00,000000000931322574182 x = x + ( x >> 30 )
+00,000000001862645147496 x = x + ( x >> 29 )
+00,000000003725290291523 x = x + ( x >> 28 )
+00,000000007450580569168 x = x + ( x >> 27 )
+00,000000014901161082825 x = x + ( x >> 26 )
+00,000000029802321943606 x = x + ( x >> 25 )
+00,000000059604642999034 x = x + ( x >> 24 )
+00,000000119209282445354 x = x + ( x >> 23 )
+00,000000238418550679858 x = x + ( x >> 22 )
+00,000000476837044516323 x = x + ( x >> 21 )
+00,000000953673861659188 x = x + ( x >> 20 )
+00,000001907346813825409 x = x + ( x >> 19 )
+00,000003814689989685890 x = x + ( x >> 18 )
+00,000007629365427567572 x = x + ( x >> 17 )
+00,000015258672648362398 x = x + ( x >> 16 )
+00,000030517112473186380 x = x + ( x >> 15 )
+00,000061033293680638527 x = x + ( x >> 14 )
+00,000122062862525677371 x = x + ( x >> 13 )
+00,000244110827527362707 x = x + ( x >> 12 )
+00,000488162079501351187 x = x + ( x >> 11 )
+00,000976085973055458925 x = x + ( x >> 10 )
+00,001951220131261749337 x = x + ( x >> 9 )
+00,003898640415657322889 x = x + ( x >> 8 )
+00,007782140442054948960 x = x + ( x >> 7 )
+00,015504186535965254479 x = x + ( x >> 6 )
+00,030771658666753687328 x = x + ( x >> 5 )
+00,060624621816434839938 x = x + ( x >> 4 )
+00,117783035656383455736 x = x + ( x >> 3 )
+00,223143551314209764858 x = x + ( x >> 2 )
+00,405465108108164384859 x = x + ( x >> 1 )
+00,693147180559945286227 x = ( x << 1 )
+01,386294361119890572454 x = ( x << 2 )
+02,772588722239781144907 x = ( x << 4 )
+05,545177444479562289814 x = ( x << 8 )
+11,090354888959124579628 x = ( x << 16 )
+21,487562597358305538364 x = ( x << 31 )
for negative values:
-00,374693449441410697531 x-= ( x >> 2 ) + ( x >> 4 )
-00,207639364778244489562 x-= ( x >> 2 ) - ( x >> 4 )
-00,115831815525121700761 x-= ( x >> 3 ) - ( x >> 6 )
-00,060380510988907482028 x-= ( x >> 4 ) - ( x >> 8 )
-00,031748698314580298119 x-= ( x >> 5 )
-00,015996403602177928366 x-= ( x >> 6 ) + ( x >> 12 )
-00,008089270731616965762 x-= ( x >> 7 ) + ( x >> 12 )
-00,004159027401785260480 x-= ( x >> 8 ) + ( x >> 12 )
-00,003423823349553609900 x-= ( x >> 8 ) - ( x >> 11 )
-00,001832733117311761669 x-= ( x >> 9 ) - ( x >> 13 )
-00,000961766059678620302 x-= ( x >> 10 ) - ( x >> 16 )
-00,000484583944571210926 x-= ( x >> 11 ) - ( x >> 18 )
-00,000243216525424976433 x-= ( x >> 12 ) - ( x >> 20 )
#endif
// input is Q16, outputQ26
static int32_t fixlog( int32_t x ) {
int32_t t,y;
// Assume an x in Q21 (shift by 15) and thus start with y with log(2^15)
y = TO_Q26(10.39720770839918);
if(x<0x00008000) x<<=16, y-= TO_Q26(11.09035488895912); /* log(65536) */
if(x<0x00800000) x<<= 8, y-= TO_Q26(5.545177444479562); /* log(256) */
if(x<0x08000000) x<<= 4, y-= TO_Q26(2.772588722239781); /* log(16) */
if(x<0x20000000) x<<= 2, y-= TO_Q26(1.386294361119891); /* log(4) */
if(x<0x40000000) x<<= 1, y-= TO_Q26(0.693147180559945); /* log(2) */
t=x+(x>>1); if((t&0x80000000)==0) x=t, y-= TO_Q26(0.405465108108164); /* log(1+1/2) */
t=x+(x>>2); if((t&0x80000000)==0) x=t, y-= TO_Q26(0.22314355131421); /* log(1+1/4) */
t=x+(x>>3); if((t&0x80000000)==0) x=t, y-= TO_Q26(0.117783035656384); /* log(1+1/8) */
t=x+(x>>4); if((t&0x80000000)==0) x=t, y-= TO_Q26(0.060624621816435); /* log(1+1/16) */
t=x+(x>>5); if((t&0x80000000)==0) x=t, y-= TO_Q26(0.030771658666754); /* log(1+1/32) */
t=x+(x>>6); if((t&0x80000000)==0) x=t, y-= TO_Q26(0.015504186535965); /* log(1+1/64) */
t=x+(x>>7); if((t&0x80000000)==0) x=t, y-= TO_Q26(0.007782140442055); /* log(1+1/128) */
t=x+(x>>8); if((t&0x80000000)==0) x=t, y-= TO_Q26(0.003898640415657); /* log(1+1/256) */
t=x+(x>>9); if((t&0x80000000)==0) x=t, y-= TO_Q26(0.001951220131262); /* log(1+1/512) */
t=x+(x>>10);if((t&0x80000000)==0) x=t, y-= TO_Q26(0.000976085973055); /* log(1+1/1024) */
t=x+(x>>11);if((t&0x80000000)==0) x=t, y-= TO_Q26(0.000488162079501); /* log(1+1/2048) */
t=x+(x>>12);if((t&0x80000000)==0) x=t, y-= TO_Q26(0.000244110827527); /* log(1+1/4096) */
t=x+(x>>13);if((t&0x80000000)==0) x=t, y-= TO_Q26(0.000122062862526); /* log(1+1/8192) */
/* Estimate residual error, log(1-x) which for small x is approx -x */
x = 2147483648-x;
/* x has been promoted to Q31, substract residual error in Q26 by shifting down by 5 */
return y - ( x >> 5 );
}
// input is Q26, output Q16
static uint32_t fixexp(int32_t x) {
int32_t t;
/* Start y with 1.0 */
uint32_t y = 0x10000;
if( x >= 0 ) {
t=x-TO_Q26(5.545177444479562); if(t>=0) x=t,y<<=8; /* log(256) */
t=x-TO_Q26(2.772588722239781); if(t>=0) x=t,y<<=4; /* log(16) */
t=x-TO_Q26(1.386294361119891); if(t>=0) x=t,y<<=2; /* log(4) */
t=x-TO_Q26(0.693147180559945); if(t>=0) x=t,y<<=1; /* log(2) */
t=x-TO_Q26(0.405465108108164); if(t>=0) x=t,y+=y>>1; /* log(1+1/2) */
t=x-TO_Q26(0.22314355131421); if(t>=0) x=t,y+=y>>2; /* log(1+1/4) */
t=x-TO_Q26(0.117783035656384); if(t>=0) x=t,y+=y>>3; /* log(1+1/8) */
t=x-TO_Q26(0.060624621816435); if(t>=0) x=t,y+=y>>4; /* log(1+1/16) */
t=x-TO_Q26(0.030771658666754); if(t>=0) x=t,y+=y>>5; /* log(1+1/32) */
t=x-TO_Q26(0.015504186535965); if(t>=0) x=t,y+=y>>6; /* log(1+1/64) */
t=x-TO_Q26(0.007782140442055); if(t>=0) x=t,y+=y>>7; /* log(1+1/128) */
t=x-TO_Q26(0.003898640415657); if(t>=0) x=t,y+=y>>8; /* log(1+1/256) */
t=x-TO_Q26(0.001951220131262); if(t>=0) x=t,y+=y>>9; /* log(1+1/512) */
t=x-TO_Q26(0.000976085973055); if(t>=0) x=t,y+=y>>10; /* log(1+1/1024) */
t=x-TO_Q26(0.000488162079501); if(t>=0) x=t,y+=y>>11; /* log(1+1/2048) */
t=x-TO_Q26(0.000244110827527); if(t>=0) x=t,y+=y>>12; /* log(1+1/4096) */
t=x-TO_Q26(0.000122062862526); if(t>=0) x=t,y+=y>>13; /* log(1+1/8192) */
/* from here the values in Q26 become approx 2^n, so lets check bits
and fix the residual error below */
if(x&0x0001000) y+=y>>14;
if(x&0x0000800) y+=y>>15;
if(x&0x0000400) y+=y>>16;
if(x&0x0000200) y+=y>>17;
if(x&0x0000100) y+=y>>18;
if(x&0x0000080) y+=y>>19;
if(x&0x0000040) y+=y>>20;
if(x&0x0000020) y+=y>>21;
if(x&0x0000010) y+=y>>22;
if(x&0x0000008) y+=y>>23;
if(x&0x0000004) y+=y>>24;
if(x&0x0000002) y+=y>>25;
if(x&0x0000001) y+=y>>26;
} else {
if(x<=TO_Q26(-11.09035488895912)) return 0;
t=x+TO_Q26(11.09035488895912); if(t<=0) x=t,y>>=16;
t=x+TO_Q26(5.545177444479562); if(t<=0) x=t,y>>=8;
t=x+TO_Q26(2.772588722239781); if(t<=0) x=t,y>>=4;
t=x+TO_Q26(1.386294361119891); if(t<=0) x=t,y>>=2;
t=x+TO_Q26(0.693147180559945); if(t<=0) x=t,y>>=1;
t=x+TO_Q26(0.374693449441411); if(t<=0) x=t,y-=(y>>4) + (y>>2);
t=x+TO_Q26(0.207639364778244); if(t<=0) x=t,y-=(y>>2) - (y>>4);
t=x+TO_Q26(0.124642445207276); if(t<=0) x=t,y-=(y>>3) - (y>>7);
t=x+TO_Q26(0.062457354933746); if(t<=0) x=t,y-=(y>>4) - (y>>9);
t=x+TO_Q26(0.031244793038107); if(t<=0) x=t,y-=(y>>5) - (y>>11);
t=x+TO_Q26(0.015748356968139); if(t<=0) x=t,y-=(y>>6);
t=x+TO_Q26(0.007966216526084); if(t<=0) x=t,y-=(y>>7) + (y>>13);
t=x+TO_Q26(0.004036455850488); if(t<=0) x=t,y-=(y>>8) + (y>>13);
t=x+TO_Q26(0.002077351513834); if(t<=0) x=t,y-=(y>>9) + (y>>13);
t=x+TO_Q26(0.001099236751907); if(t<=0) x=t,y-=(y>>10)+ (y>>13);
t=x+TO_Q26(0.000610537902840); if(t<=0) x=t,y-=(y>>11)+ (y>>13);
t=x+TO_Q26(0.000366278009100); if(t<=0) x=t,y-=(y>>12)+ (y>>13);
t=x+TO_Q26(0.000213645867528); if(t<=0) x=t,y-=(y>>12)- (y>>15);
if(x&0x0002000) y-=y>>13;
if(x&0x0001000) y-=y>>14;
if(x&0x0000800) y-=y>>15;
if(x&0x0000400) y-=y>>16;
if(x&0x0000200) y-=y>>17;
if(x&0x0000100) y-=y>>18;
if(x&0x0000080) y-=y>>19;
if(x&0x0000040) y-=y>>20;
if(x&0x0000020) y-=y>>21;
if(x&0x0000010) y-=y>>22;
if(x&0x0000008) y-=y>>23;
if(x&0x0000004) y-=y>>24;
if(x&0x0000002) y-=y>>25;
if(x&0x0000001) y-=y>>26;
}
return y;
}
int fixpow(int32_t *pow, int32_t base, int32_t exponent )
{
int32_t log_base, res;
int64_t log_base_times_exponent;
int neg = 0;
if( base < 0 ) {
/* negative bases only can have integer exponents */
if( exponent & 0xffff ) return -1;
/* sign of power of a negative base is determined by
wether exp is even */
if( exponent & 0x10000 ) neg = 1;
base = abs(base);
}
/* To calculate pow(base,exp), we do exp( log(base) * exp )
which is mathematically the same. log_base is Q26 */
log_base = fixlog( base );
log_base_times_exponent = ( (int64_t)log_base * (int64_t)exponent ) >> 16;
/* In Q0, fixexp overflows for values > 21,48756259689264 which
in Q26 notation is 1442005916 */
if( log_base_times_exponent > 1442005916 )
return -2;
/* In Q16, fixexp overflows for values > 10,39720770839918 */
if( log_base_times_exponent > TO_Q26(10.39720770839918) )
return -2;
res = (int32_t)log_base_times_exponent;
res = fixexp( res );
if( neg ) res = -res;
*pow = res;
return 0;
}
int main()
{
double base = -.5;
double exponent = 10.;
int32_t result, input, output;
int error = fixpow( &result, TO_Q16(base), TO_Q16(exponent) );
// Testing fixlog
for (input=1; input<=INT32_MAX/2; ++input) {
int32_t expected = TO_Q26(log(FROM_Q16(input)));
int32_t output = fixlog(input);
if( abs(output-expected) > 8 )
printf( "%08X : %08X = %08X (%08X)\n", output, expected, abs(output-expected), input );
}
// Testing fixexp
for (input=-2147483648; input < TO_Q26(10.39720770839918); ++input ) {
double expected = exp(FROM_Q26(input));
int32_t expected_int = TO_Q16(expected);
int32_t output_int = fixexp(input);
double output = FROM_Q16(output_int);
if( abs(output_int-expected_int) > 8 ) {
if( expected > output && output > 0 && expected / output > 1.002 )
printf( "%08X : %08X = %08X (%08d)\n", output_int, expected_int, abs(output_int-expected_int), input );
if( expected < output && expected > 0 && output / expected > 1.002 )
printf( "%08X : %08X = %08X (%08d)\n", output_int, expected_int, abs(output_int-expected_int), input );
}
}
printf( "pow(%lf,%lf)=%lf (%s)\n", base, exponent, FROM_Q16(result), error?"ERROR":"OK" );
return 0;
}
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