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Fixed point utility: implemented fixed_resolution_shift of 16 and 24

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f4exb 2018-02-04 01:25:09 +01:00
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461
sdrbase/util/fixed.cpp
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@ -1,461 +0,0 @@
// Distributed under the Boost Software License, Version 1.0. (See
// accompanying file LICENSE_1_0.txt or copy at
// http://www.boost.org/LICENSE_1_0.txt)
// (C) Copyright 2007 Anthony Williams
#include "fixed.hpp"
// all constants are scaled to 2^28 (as 1)
int64_t const internal_pi=0x3243f6a8;
int64_t const internal_two_pi=0x6487ed51;
int64_t const internal_half_pi=0x1921fb54;
int64_t const internal_quarter_pi=0xc90fdaa;
extern fixed const fixed_pi(fixed::internal(),internal_pi);
extern fixed const fixed_two_pi(fixed::internal(),internal_two_pi);
extern fixed const fixed_half_pi(fixed::internal(),internal_half_pi);
extern fixed const fixed_quarter_pi(fixed::internal(),internal_quarter_pi);
fixed& fixed::operator%=(fixed const& other)
{
m_nVal = m_nVal%other.m_nVal;
return *this;
}
fixed& fixed::operator*=(fixed const& val)
{
bool const val_negative=val.m_nVal<0;
bool const this_negative=m_nVal<0;
bool const negate=val_negative ^ this_negative;
uint64_t const other=val_negative?-val.m_nVal:val.m_nVal;
uint64_t const self=this_negative?-m_nVal:m_nVal;
if(uint64_t const self_upper=(self>>32))
{
m_nVal=(self_upper*other)<<(32-fixed_resolution_shift);
}
else
{
m_nVal=0;
}
if(uint64_t const self_lower=(self&0xffffffff))
{
uint64_t const other_upper=static_cast<uint64_t>(other>>32);
uint64_t const other_lower=static_cast<uint64_t>(other&0xffffffff);
uint64_t const lower_self_upper_other_res=self_lower*other_upper;
uint64_t const lower_self_lower_other_res=self_lower*other_lower;
m_nVal+=(lower_self_upper_other_res<<(32-fixed_resolution_shift))
+ (lower_self_lower_other_res>>fixed_resolution_shift);
}
if(negate)
{
m_nVal=-m_nVal;
}
return *this;
}
fixed& fixed::operator/=(fixed const& divisor)
{
if( !divisor.m_nVal)
{
m_nVal=fixed_max.m_nVal;
}
else
{
bool const negate_this=(m_nVal<0);
bool const negate_divisor=(divisor.m_nVal<0);
bool const negate=negate_this ^ negate_divisor;
uint64_t a=negate_this?-m_nVal:m_nVal;
uint64_t b=negate_divisor?-divisor.m_nVal:divisor.m_nVal;
uint64_t res=0;
uint64_t temp=b;
bool const a_large=a>b;
unsigned shift=fixed_resolution_shift;
if(a_large)
{
uint64_t const half_a=a>>1;
while(temp<half_a)
{
temp<<=1;
++shift;
}
}
uint64_t d=1LL<<shift;
if(a_large)
{
a-=temp;
res+=d;
}
while(a && temp && shift)
{
unsigned right_shift=0;
while(right_shift<shift && (temp>a))
{
temp>>=1;
++right_shift;
}
d>>=right_shift;
shift-=right_shift;
a-=temp;
res+=d;
}
m_nVal=(negate?-(int64_t)res:res);
}
return *this;
}
fixed fixed::sqrt() const
{
unsigned const max_shift=62;
uint64_t a_squared=1LL<<max_shift;
unsigned b_shift=(max_shift+fixed_resolution_shift)/2;
uint64_t a=1LL<<b_shift;
uint64_t x=m_nVal;
while(b_shift && a_squared>x)
{
a>>=1;
a_squared>>=2;
--b_shift;
}
uint64_t remainder=x-a_squared;
--b_shift;
while(remainder && b_shift)
{
uint64_t b_squared=1LL<<(2*b_shift-fixed_resolution_shift);
int const two_a_b_shift=b_shift+1-fixed_resolution_shift;
uint64_t two_a_b=(two_a_b_shift>0)?(a<<two_a_b_shift):(a>>-two_a_b_shift);
while(b_shift && remainder<(b_squared+two_a_b))
{
b_squared>>=2;
two_a_b>>=1;
--b_shift;
}
uint64_t const delta=b_squared+two_a_b;
if((2*remainder)>delta)
{
a+=(1LL<<b_shift);
remainder-=delta;
if(b_shift)
{
--b_shift;
}
}
}
return fixed(internal(),a);
}
namespace
{
int const max_power=63-fixed_resolution_shift;
// 35 values (63-28)
int64_t const log_two_power_n_reversed[]={
0x18429946ELL,0x1791272EFLL,0x16DFB516FLL,0x162E42FF0LL,0x157CD0E70LL,0x14CB5ECF1LL,0x1419ECB71LL,0x13687A9F2LL,
0x12B708872LL,0x1205966F3LL,0x115424573LL,0x10A2B23F4LL,0xFF140274LL,0xF3FCE0F5LL,0xE8E5BF75LL,0xDDCE9DF6LL,
0xD2B77C76LL,0xC7A05AF7LL,0xBC893977LL,0xB17217F8LL,0xA65AF679LL,0x9B43D4F9LL,0x902CB379LL,0x851591FaLL,
0x79FE707bLL,0x6EE74EFbLL,0x63D02D7BLL,0x58B90BFcLL,0x4DA1EA7CLL,0x428AC8FdLL,0x3773A77DLL,0x2C5C85FeLL,
0x2145647ELL,0x162E42FfLL,0xB17217FLL
};
// 28 values
int64_t const log_one_plus_two_power_minus_n[]={
0x67CC8FBLL,0x391FEF9LL,0x1E27077LL,0xF85186LL,
0x7E0A6CLL,0x3F8151LL,0x1FE02ALL,0xFF805LL,0x7FE01LL,0x3FF80LL,0x1FFE0LL,0xFFF8LL,
0x7FFELL,0x4000LL,0x2000LL,0x1000LL,0x800LL,0x400LL,0x200LL,0x100LL,
0x80LL,0x40LL,0x20LL,0x10LL,0x8LL,0x4LL,0x2LL,0x1LL
};
// 28 values
int64_t const log_one_over_one_minus_two_power_minus_n[]={
0xB172180LL,0x49A5884LL,0x222F1D0LL,0x108598BLL,
0x820AECLL,0x408159LL,0x20202BLL,0x100805LL,0x80201LL,0x40080LL,0x20020LL,0x10008LL,
0x8002LL,0x4001LL,0x2000LL,0x1000LL,0x800LL,0x400LL,0x200LL,0x100LL,
0x80LL,0x40LL,0x20LL,0x10LL,0x8LL,0x4LL,0x2LL,0x1LL
};
}
fixed fixed::exp() const
{
if(m_nVal>=log_two_power_n_reversed[0])
{
return fixed_max;
}
if(m_nVal<-log_two_power_n_reversed[63-2*fixed_resolution_shift])
{
return fixed(internal(),0);
}
if(!m_nVal)
{
return fixed(internal(),fixed_resolution);
}
int64_t res=fixed_resolution;
if(m_nVal>0)
{
int power=max_power;
int64_t const* log_entry=log_two_power_n_reversed;
int64_t temp=m_nVal;
while(temp && power>(-(int)fixed_resolution_shift))
{
while(!power || (temp<*log_entry))
{
if(!power)
{
log_entry=log_one_plus_two_power_minus_n;
}
else
{
++log_entry;
}
--power;
}
temp-=*log_entry;
if(power<0)
{
res+=(res>>(-power));
}
else
{
res<<=power;
}
}
}
else
{
int power=fixed_resolution_shift;
int64_t const* log_entry=log_two_power_n_reversed+(max_power-power);
int64_t temp=m_nVal;
while(temp && power>(-(int)fixed_resolution_shift))
{
while(!power || (temp>(-*log_entry)))
{
if(!power)
{
log_entry=log_one_over_one_minus_two_power_minus_n;
}
else
{
++log_entry;
}
--power;
}
temp+=*log_entry;
if(power<0)
{
res-=(res>>(-power));
}
else
{
res>>=power;
}
}
}
return fixed(internal(),res);
}
fixed fixed::log() const
{
if(m_nVal<=0)
{
return -fixed_max;
}
if(m_nVal==fixed_resolution)
{
return fixed_zero;
}
uint64_t temp=m_nVal;
int left_shift=0;
uint64_t const scale_position=0x8000000000000000;
while(temp<scale_position)
{
++left_shift;
temp<<=1;
}
int64_t res=(left_shift<max_power)?
log_two_power_n_reversed[left_shift]:
-log_two_power_n_reversed[2*max_power-left_shift];
unsigned right_shift=1;
uint64_t shifted_temp=temp>>1;
while(temp && (right_shift<fixed_resolution_shift))
{
while((right_shift<fixed_resolution_shift) && (temp<(shifted_temp+scale_position)))
{
shifted_temp>>=1;
++right_shift;
}
temp-=shifted_temp;
shifted_temp=temp>>right_shift;
res+=log_one_over_one_minus_two_power_minus_n[right_shift-1];
}
return fixed(fixed::internal(),res);
}
namespace
{
const int64_t arctantab[32] = {
297197971, 210828714, 124459457, 65760959, 33381290, 16755422, 8385879,
4193963, 2097109, 1048571, 524287, 262144, 131072, 65536, 32768, 16384,
8192, 4096, 2048, 1024, 512, 256, 128, 64, 32, 16, 8, 4, 2, 1, 0, 0,
};
int64_t scale_cordic_result(int64_t a)
{
int64_t const cordic_scale_factor=0x22C2DD1C; /* 0.271572 * 2^31*/
return (int64_t)((((int64_t)a)*cordic_scale_factor)>>31);
}
int64_t right_shift(int64_t val,int shift)
{
return (shift<0)?(val<<-shift):(val>>shift);
}
void perform_cordic_rotation(int64_t&px, int64_t&py, int64_t theta)
{
int64_t x = px, y = py;
int64_t const *arctanptr = arctantab;
for (int i = -1; i <= (int)fixed_resolution_shift; ++i)
{
int64_t const yshift=right_shift(y,i);
int64_t const xshift=right_shift(x,i);
if (theta < 0)
{
x += yshift;
y -= xshift;
theta += *arctanptr++;
}
else
{
x -= yshift;
y += xshift;
theta -= *arctanptr++;
}
}
px = scale_cordic_result(x);
py = scale_cordic_result(y);
}
void perform_cordic_polarization(int64_t& argx, int64_t&argy)
{
int64_t theta=0;
int64_t x = argx, y = argy;
int64_t const *arctanptr = arctantab;
for(int i = -1; i <= (int)fixed_resolution_shift; ++i)
{
int64_t const yshift=right_shift(y,i);
int64_t const xshift=right_shift(x,i);
if(y < 0)
{
y += xshift;
x -= yshift;
theta -= *arctanptr++;
}
else
{
y -= xshift;
x += yshift;
theta += *arctanptr++;
}
}
argx = scale_cordic_result(x);
argy = theta;
}
}
void fixed::sin_cos(fixed const& theta,fixed* s,fixed*c)
{
int64_t x=theta.m_nVal%internal_two_pi;
if( x < 0 )
x += internal_two_pi;
bool negate_cos=false;
bool negate_sin=false;
if( x > internal_pi )
{
x =internal_two_pi-x;
negate_sin=true;
}
if(x>internal_half_pi)
{
x=internal_pi-x;
negate_cos=true;
}
int64_t x_cos=1<<28,x_sin=0;
perform_cordic_rotation(x_cos,x_sin,(int64_t)x);
if(s)
{
s->m_nVal=negate_sin?-x_sin:x_sin;
}
if(c)
{
c->m_nVal=negate_cos?-x_cos:x_cos;
}
}
fixed fixed::atan() const
{
fixed r,theta;
to_polar(1,*this,&r,&theta);
return theta;
}
void fixed::to_polar(fixed const& x,fixed const& y,fixed* r,fixed*theta)
{
bool const negative_x=x.m_nVal<0;
bool const negative_y=y.m_nVal<0;
uint64_t a=negative_x?-x.m_nVal:x.m_nVal;
uint64_t b=negative_y?-y.m_nVal:y.m_nVal;
unsigned right_shift=0;
unsigned const max_value=1U<<fixed_resolution_shift;
while((a>=max_value) || (b>=max_value))
{
++right_shift;
a>>=1;
b>>=1;
}
int64_t xtemp=(int64_t)a;
int64_t ytemp=(int64_t)b;
perform_cordic_polarization(xtemp,ytemp);
r->m_nVal=int64_t(xtemp)<<right_shift;
theta->m_nVal=ytemp;
if(negative_x && negative_y)
{
theta->m_nVal-=internal_pi;
}
else if(negative_x)
{
theta->m_nVal=internal_pi-theta->m_nVal;
}
else if(negative_y)
{
theta->m_nVal=-theta->m_nVal;
}
}

9
sdrbase/util/fixed.h
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@ -4,6 +4,9 @@
// Distributed under the Boost Software License, Version 1.0. //
// See: http://www.boost.org/LICENSE_1_0.txt) //
// //
// Original article: //
// http://www.drdobbs.com/cpp/optimizing-math-intensive-applications-w/207000448 //
// //
// Copyright (C) 2018 Edouard Griffiths, F4EXB //
// //
// Modified as fully templatized class with variable size and type internal //
@ -740,7 +743,7 @@ void Fixed<IntType, IntBits>::sin_cos(Fixed<IntType, IntBits> const& theta, Fixe
negate_cos=true;
}
int64_t x_cos = 1<<28, x_sin = 0;
int64_t x_cos = 1<<FixedTraits<IntBits>::fixed_resolution_shift, x_sin = 0;
perform_cordic_rotation(x_cos, x_sin, (int64_t) x);
@ -2347,7 +2350,7 @@ template<typename IntType, uint32_t IntBits>
inline Fixed<IntType, IntBits> Fixed<IntType, IntBits>::sin() const
{
Fixed res;
sin_cos(*this,&res, 0);
sin_cos(*this, &res, 0);
return res;
}
@ -2355,7 +2358,7 @@ template<typename IntType, uint32_t IntBits>
inline Fixed<IntType, IntBits> Fixed<IntType, IntBits>::cos() const
{
Fixed res;
sin_cos(*this,0, &res);
sin_cos(*this, 0, &res);
return res;
}

1591
sdrbase/util/fixed.hpp
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Plik diff jest za duży Load Diff

403
sdrbase/util/fixedtraits.cpp
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@ -17,30 +17,401 @@
#include "fixedtraits.h"
// 1.0 = 2^28 internal representation
// ln(1/2^n), n = index+1 then list is reversed
const int64_t FixedTraits<28>::log_two_power_n_reversed[35] = {
0x18429946ELL,0x1791272EFLL,0x16DFB516FLL,0x162E42FF0LL,0x157CD0E70LL,0x14CB5ECF1LL,0x1419ECB71LL,0x13687A9F2LL,
0x12B708872LL,0x1205966F3LL,0x115424573LL,0x10A2B23F4LL,0xFF140274LL,0xF3FCE0F5LL,0xE8E5BF75LL,0xDDCE9DF6LL,
0xD2B77C76LL,0xC7A05AF7LL,0xBC893977LL,0xB17217F8LL,0xA65AF679LL,0x9B43D4F9LL,0x902CB379LL,0x851591FaLL,
0x79FE707bLL,0x6EE74EFbLL,0x63D02D7BLL,0x58B90BFcLL,0x4DA1EA7CLL,0x428AC8FdLL,0x3773A77DLL,0x2C5C85FeLL,
0x2145647ELL,0x162E42FfLL,0xB17217FLL
0x18429946ELL,
0x1791272EFLL,
0x16DFB516FLL,
0x162E42FF0LL,
0x157CD0E70LL,
0x14CB5ECF1LL,
0x1419ECB71LL,
0x13687A9F2LL,
0x12B708872LL,
0x1205966F3LL,
0x115424573LL,
0x10A2B23F4LL,
0xFF140274LL,
0xF3FCE0F5LL,
0xE8E5BF75LL,
0xDDCE9DF6LL,
0xD2B77C76LL,
0xC7A05AF7LL,
0xBC893977LL,
0xB17217F8LL,
0xA65AF679LL,
0x9B43D4F9LL,
0x902CB379LL,
0x851591FaLL,
0x79FE707bLL,
0x6EE74EFbLL,
0x63D02D7BLL,
0x58B90BFcLL,
0x4DA1EA7CLL,
0x428AC8FdLL,
0x3773A77DLL,
0x2C5C85FeLL,
0x2145647ELL,
0x162E42FfLL,
0xB17217FLL
};
// ln(1+2^-n), n = index+1
const int64_t FixedTraits<28>::log_one_plus_two_power_minus_n[28] = {
0x67CC8FBLL,0x391FEF9LL,0x1E27077LL,0xF85186LL,
0x7E0A6CLL,0x3F8151LL,0x1FE02ALL,0xFF805LL,0x7FE01LL,0x3FF80LL,0x1FFE0LL,0xFFF8LL,
0x7FFELL,0x4000LL,0x2000LL,0x1000LL,0x800LL,0x400LL,0x200LL,0x100LL,
0x80LL,0x40LL,0x20LL,0x10LL,0x8LL,0x4LL,0x2LL,0x1LL
0x67CC8FBLL,
0x391FEF9LL,
0x1E27077LL,
0xF85186LL,
0x7E0A6CLL,
0x3F8151LL,
0x1FE02ALL,
0xFF805LL,
0x7FE01LL,
0x3FF80LL,
0x1FFE0LL,
0xFFF8LL,
0x7FFELL,
0x4000LL,
0x2000LL,
0x1000LL,
0x800LL,
0x400LL,
0x200LL,
0x100LL,
0x80LL,
0x40LL,
0x20LL,
0x10LL,
0x8LL,
0x4LL,
0x2LL,
0x1LL
};
// ln(1/1-2^-n), n = index+1
const int64_t FixedTraits<28>::log_one_over_one_minus_two_power_minus_n[28] = {
0xB172180LL,0x49A5884LL,0x222F1D0LL,0x108598BLL,
0x820AECLL,0x408159LL,0x20202BLL,0x100805LL,0x80201LL,0x40080LL,0x20020LL,0x10008LL,
0x8002LL,0x4001LL,0x2000LL,0x1000LL,0x800LL,0x400LL,0x200LL,0x100LL,
0x80LL,0x40LL,0x20LL,0x10LL,0x8LL,0x4LL,0x2LL,0x1LL
0xB172180LL,
0x49A5884LL,
0x222F1D0LL,
0x108598BLL,
0x820AECLL,
0x408159LL,
0x20202BLL,
0x100805LL,
0x80201LL,
0x40080LL,
0x20020LL,
0x10008LL,
0x8002LL,
0x4001LL,
0x2000LL,
0x1000LL,
0x800LL,
0x400LL,
0x200LL,
0x100LL,
0x80LL,
0x40LL,
0x20LL,
0x10LL,
0x8LL,
0x4LL,
0x2LL,
0x1LL
};
const int64_t FixedTraits<28>::arctantab[32] = {
297197971, 210828714, 124459457, 65760959, 33381290, 16755422, 8385879,
4193963, 2097109, 1048571, 524287, 262144, 131072, 65536, 32768, 16384,
8192, 4096, 2048, 1024, 512, 256, 128, 64, 32, 16, 8, 4, 2, 1, 0, 0,
297197971,
210828714,
124459457,
65760959,
33381290,
16755422,
8385879,
4193963,
2097109,
1048571,
524287,
262144,
131072,
65536,
32768,
16384,
8192,
4096,
2048,
1024,
512,
256,
128,
64,
32,
16,
8,
4,
2,
1,
0,
0
};
// 1.0 = 2^16 internal representation
const int64_t FixedTraits<16>::log_two_power_n_reversed[47] = {
2135026LL,
2089600LL,
2044174LL,
1998748LL,
1953322LL,
1907896LL,
1862470LL,
1817044LL,
1771618LL,
1726192LL,
1680765LL,
1635339LL,
1589913LL,
1544487LL,
1499061LL,
1453635LL,
1408209LL,
1362783LL,
1317357LL,
1271931LL,
1226505LL,
1181078LL,
1135652LL,
1090226LL,
1044800LL,
999374LL,
953948LL,
908522LL,
863096LL,
817670LL,
772244LL,
726817LL,
681391LL,
635965LL,
590539LL,
545113LL,
499687LL,
454261LL,
408835LL,
363409LL,
317983LL,
272557LL,
227130LL,
181704LL,
136278LL,
90852LL,
45426LL
};
const int64_t FixedTraits<16>::log_one_plus_two_power_minus_n[16] = {
26573LL,
14624LL,
7719LL,
3973LL,
2017LL,
1016LL,
510LL,
256LL,
128LL,
64LL,
32LL,
16LL,
8LL,
4LL,
2LL,
1LL
};
const int64_t FixedTraits<16>::log_one_over_one_minus_two_power_minus_n[16] = {
45426LL,
18854LL,
8751LL,
4230LL,
2081LL,
1032LL,
514LL,
257LL,
128LL,
64LL,
32LL,
16LL,
8LL,
4LL,
2LL,
1LL
};
const int64_t FixedTraits<16>::arctantab[32] = {
72558LL,
51471LL,
30385LL,
16054LL,
8149LL,
4090LL,
2047LL,
1023LL,
511LL,
255LL,
127LL,
64LL,
32LL,
16LL,
8LL,
4LL,
2LL,
1LL,
0LL,
0LL,
0LL,
0LL,
0LL,
0LL,
0LL,
0LL,
0LL,
0LL,
0LL,
0LL,
0LL,
0LL
};
// 1.0 = 2^24 internal representation
const int64_t FixedTraits<24>::log_two_power_n_reversed[39] = {
453534119LL,
441905039LL,
430275959LL,
418646879LL,
407017799LL,
395388719LL,
383759639LL,
372130559LL,
360501479LL,
348872399LL,
337243319LL,
325614239LL,
313985159LL,
302356079LL,
290726999LL,
279097919LL,
267468839LL,
255839759LL,
244210679LL,
232581599LL,
220952519LL,
209323439LL,
197694359LL,
186065279LL,
174436200LL,
162807120LL,
151178040LL,
139548960LL,
127919880LL,
116290800LL,
104661720LL,
93032640LL,
81403560LL,
69774480LL,
58145400LL,
46516320LL,
34887240LL,
23258160LL,
11629080LL
};
const int64_t FixedTraits<24>::log_one_plus_two_power_minus_n[24] = {
6802576LL,
3743728LL,
1976071LL,
1017112LL,
516263LL,
260117LL,
130563LL,
65408LL,
32736LL,
16376LL,
8190LL,
4096LL,
2048LL,
1024LL,
512LL,
256LL,
128LL,
64LL,
32LL,
16LL,
8LL,
4LL,
2LL,
1LL
};
const int64_t FixedTraits<24>::log_one_over_one_minus_two_power_minus_n[24] = {
11629080LL,
4826504LL,
2240285LL,
1082777LL,
532655LL,
264214LL,
131587LL,
65664LL,
32800LL,
16392LL,
8194LL,
4097LL,
2048LL,
1024LL,
512LL,
256LL,
128LL,
64LL,
32LL,
16LL,
8LL,
4LL,
2LL,
1LL
};
const int64_t FixedTraits<24>::arctantab[32] = {
18574873LL,
13176794LL,
7778716LL,
4110059LL,
2086330LL,
1047213LL,
524117LL,
262122LL,
131069LL,
65535LL,
32767LL,
16384LL,
8192LL,
4096LL,
2048LL,
1024LL,
512LL,
256LL,
128LL,
64LL,
32LL,
16LL,
8LL,
4LL,
2LL,
1LL,
0LL,
0LL,
0LL,
0LL,
0LL,
0LL
};

32
sdrbase/util/fixedtraits.h
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@ -41,4 +41,36 @@ struct FixedTraits<28>
static const int64_t arctantab[32];
};
template<>
struct FixedTraits<16>
{
static const uint32_t fixed_resolution_shift = 16;
static const int64_t fixed_resolution = 1LL << fixed_resolution_shift;
static const int32_t max_power = 63 - fixed_resolution_shift;
static const int64_t internal_pi = 205887;
static const int64_t internal_two_pi = 411775;
static const int64_t internal_half_pi = 102944;
static const int64_t internal_quarter_pi = 51472;
static const int64_t log_two_power_n_reversed[47]; // 47 = 63 - 16
static const int64_t log_one_plus_two_power_minus_n[16];
static const int64_t log_one_over_one_minus_two_power_minus_n[16];
static const int64_t arctantab[32];
};
template<>
struct FixedTraits<24>
{
static const uint32_t fixed_resolution_shift = 24;
static const int64_t fixed_resolution = 1LL << fixed_resolution_shift;
static const int32_t max_power = 63 - fixed_resolution_shift;
static const int64_t internal_pi = 52707179;
static const int64_t internal_two_pi = 105414357;
static const int64_t internal_half_pi = 26353589;
static const int64_t internal_quarter_pi = 13176795;
static const int64_t log_two_power_n_reversed[39]; // 39 = 63 - 16
static const int64_t log_one_plus_two_power_minus_n[24];
static const int64_t log_one_over_one_minus_two_power_minus_n[24];
static const int64_t arctantab[32];
};
#endif /* SDRBASE_UTIL_FIXEDTRAITS_H_ */