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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 ) {
    int32_t expected = TO_Q16(exp(FROM_Q26(input)));
    int32_t output   = fixexp(input);
    if( abs(output-expected) > 16 )
        printf( "%08X : %08X = %08X (%08d)\n", output, expected, abs(output-expected), input );
  }

  printf( "pow(%lf,%lf)=%lf (%s)\n", base, exponent, FROM_Q16(result), error?"ERROR":"OK" );
  return 0;
}