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Mini Shell
Mini Shell
/***************************************************************************
* Copyright (c) Johan Mabille, Sylvain Corlay, Wolf Vollprecht and *
* Martin Renou *
* Copyright (c) QuantStack *
* *
* Distributed under the terms of the BSD 3-Clause License. *
* *
* The full license is in the file LICENSE, distributed with this software. *
****************************************************************************/
#include <cmath>
#include <cstdint>
#include <cstring>
namespace xsimd
{
namespace detail
{
/* origin: boost/simd/arch/common/scalar/function/rem_pio2.hpp */
/*
* ====================================================
* copyright 2016 NumScale SAS
*
* Distributed under the Boost Software License, Version 1.0.
* (See copy at http://boost.org/LICENSE_1_0.txt)
* ====================================================
*/
#if defined(_MSC_VER)
#define ONCE0 \
__pragma(warning(push)) \
__pragma(warning(disable : 4127)) while (0) \
__pragma(warning(pop)) /**/
#else
#define ONCE0 while (0)
#endif
/*
* ====================================================
* Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved.
*
* Developed at SunPro, a Sun Microsystems, Inc. business.
* Permission to use, copy, modify, and distribute this
* software is freely granted, provided that this notice
* is preserved.
* ====================================================
*/
#if defined(__GNUC__) && defined(__BYTE_ORDER__)
#if __BYTE_ORDER__ == __ORDER_LITTLE_ENDIAN__
#define XSIMD_LITTLE_ENDIAN
#endif
#elif defined(_WIN32)
// We can safely assume that Windows is always little endian
#define XSIMD_LITTLE_ENDIAN
#elif defined(i386) || defined(i486) || \
defined(intel) || defined(x86) || defined(i86pc) || \
defined(__alpha) || defined(__osf__)
#define XSIMD_LITTLE_ENDIAN
#endif
#ifdef XSIMD_LITTLE_ENDIAN
#define LOW_WORD_IDX 0
#define HIGH_WORD_IDX sizeof(std::uint32_t)
#else
#define LOW_WORD_IDX sizeof(std::uint32_t)
#define HIGH_WORD_IDX 0
#endif
#define GET_HIGH_WORD(i, d) \
do \
{ \
double f = (d); \
std::memcpy(&(i), reinterpret_cast<char*>(&f) + \
HIGH_WORD_IDX, \
sizeof(std::uint32_t)); \
} \
ONCE0 \
/**/
#define GET_LOW_WORD(i, d) \
do \
{ \
double f = (d); \
std::memcpy(&(i), reinterpret_cast<char*>(&f) + \
LOW_WORD_IDX, \
sizeof(std::uint32_t)); \
} \
ONCE0 \
/**/
#define SET_HIGH_WORD(d, v) \
do \
{ \
double f = (d); \
std::uint32_t value = (v); \
std::memcpy(reinterpret_cast<char*>(&f) + \
HIGH_WORD_IDX, \
&value, sizeof(std::uint32_t)); \
(d) = f; \
} \
ONCE0 \
/**/
#define SET_LOW_WORD(d, v) \
do \
{ \
double f = (d); \
std::uint32_t value = (v); \
std::memcpy(reinterpret_cast<char*>(&f) + \
LOW_WORD_IDX, \
&value, sizeof(std::uint32_t)); \
(d) = f; \
} \
ONCE0 \
/**/
/*
* __kernel_rem_pio2(x,y,e0,nx,prec,ipio2)
* double x[],y[]; int e0,nx,prec; int ipio2[];
*
* __kernel_rem_pio2 return the last three digits of N with
* y = x - N*pi/2
* so that |y| < pi/2.
*
* The method is to compute the integer (mod 8) and fraction parts of
* (2/pi)*x without doing the full multiplication. In general we
* skip the part of the product that are known to be a huge integer (
* more accurately, = 0 mod 8 ). Thus the number of operations are
* independent of the exponent of the input.
*
* (2/pi) is represented by an array of 24-bit integers in ipio2[].
*
* Input parameters:
* x[] The input value (must be positive) is broken into nx
* pieces of 24-bit integers in double precision format.
* x[i] will be the i-th 24 bit of x. The scaled exponent
* of x[0] is given in input parameter e0 (i.e., x[0]*2^e0
* match x's up to 24 bits.
*
* Example of breaking a double positive z into x[0]+x[1]+x[2]:
* e0 = ilogb(z)-23
* z = scalbn(z,-e0)
* for i = 0,1,2
* x[i] = floor(z)
* z = (z-x[i])*2**24
*
*
* y[] ouput result in an array of double precision numbers.
* The dimension of y[] is:
* 24-bit precision 1
* 53-bit precision 2
* 64-bit precision 2
* 113-bit precision 3
* The actual value is the sum of them. Thus for 113-bit
* precison, one may have to do something like:
*
* long double t,w,r_head, r_tail;
* t = (long double)y[2] + (long double)y[1];
* w = (long double)y[0];
* r_head = t+w;
* r_tail = w - (r_head - t);
*
* e0 The exponent of x[0]
*
* nx dimension of x[]
*
* prec an integer indicating the precision:
* 0 24 bits (single)
* 1 53 bits (double)
* 2 64 bits (extended)
* 3 113 bits (quad)
*
* ipio2[]
* integer array, contains the (24*i)-th to (24*i+23)-th
* bit of 2/pi after binary point. The corresponding
* floating value is
*
* ipio2[i] * 2^(-24(i+1)).
*
* External function:
* double scalbn(), floor();
*
*
* Here is the description of some local variables:
*
* jk jk+1 is the initial number of terms of ipio2[] needed
* in the computation. The recommended value is 2,3,4,
* 6 for single, double, extended,and quad.
*
* jz local integer variable indicating the number of
* terms of ipio2[] used.
*
* jx nx - 1
*
* jv index for pointing to the suitable ipio2[] for the
* computation. In general, we want
* ( 2^e0*x[0] * ipio2[jv-1]*2^(-24jv) )/8
* is an integer. Thus
* e0-3-24*jv >= 0 or (e0-3)/24 >= jv
* Hence jv = max(0,(e0-3)/24).
*
* jp jp+1 is the number of terms in PIo2[] needed, jp = jk.
*
* q[] double array with integral value, representing the
* 24-bits chunk of the product of x and 2/pi.
*
* q0 the corresponding exponent of q[0]. Note that the
* exponent for q[i] would be q0-24*i.
*
* PIo2[] double precision array, obtained by cutting pi/2
* into 24 bits chunks.
*
* f[] ipio2[] in floating point
*
* iq[] integer array by breaking up q[] in 24-bits chunk.
*
* fq[] final product of x*(2/pi) in fq[0],..,fq[jk]
*
* ih integer. If >0 it indicates q[] is >= 0.5, hence
* it also indicates the *sign* of the result.
*
*/
inline int32_t __kernel_rem_pio2(double* x, double* y, int32_t e0, int32_t nx, int32_t prec, const int32_t* ipio2)
{
static const int32_t init_jk[] = {2, 3, 4, 6}; /* initial value for jk */
static const double PIo2[] = {
1.57079625129699707031e+00, /* 0x3FF921FB, 0x40000000 */
7.54978941586159635335e-08, /* 0x3E74442D, 0x00000000 */
5.39030252995776476554e-15, /* 0x3CF84698, 0x80000000 */
3.28200341580791294123e-22, /* 0x3B78CC51, 0x60000000 */
1.27065575308067607349e-29, /* 0x39F01B83, 0x80000000 */
1.22933308981111328932e-36, /* 0x387A2520, 0x40000000 */
2.73370053816464559624e-44, /* 0x36E38222, 0x80000000 */
2.16741683877804819444e-51, /* 0x3569F31D, 0x00000000 */
};
static const double
zero = 0.0,
one = 1.0,
two24 = 1.67772160000000000000e+07, /* 0x41700000, 0x00000000 */
twon24 = 5.96046447753906250000e-08; /* 0x3E700000, 0x00000000 */
int32_t jz, jx, jv, jp, jk, carry, n, iq[20], i, j, k, m, q0, ih;
double z, fw, f[20], fq[20], q[20];
/* initialize jk*/
jk = init_jk[prec];
jp = jk;
/* determine jx,jv,q0, note that 3>q0 */
jx = nx - 1;
jv = (e0 - 3) / 24;
if (jv < 0)
jv = 0;
q0 = e0 - 24 * (jv + 1);
/* set up f[0] to f[jx+jk] where f[jx+jk] = ipio2[jv+jk] */
j = jv - jx;
m = jx + jk;
for (i = 0; i <= m; i++, j++)
f[i] = (j < 0) ? zero : (double)ipio2[j];
/* compute q[0],q[1],...q[jk] */
for (i = 0; i <= jk; i++)
{
for (j = 0, fw = 0.0; j <= jx; j++)
fw += x[j] * f[jx + i - j];
q[i] = fw;
}
jz = jk;
recompute:
/* distill q[] into iq[] reversingly */
for (i = 0, j = jz, z = q[jz]; j > 0; i++, j--)
{
fw = (double)((int32_t)(twon24 * z));
iq[i] = (int)(z - two24 * fw);
z = q[j - 1] + fw;
}
/* compute n */
z = std::scalbn(z, q0); /* actual value of z */
z -= 8.0 * std::floor(z * 0.125); /* trim off integer >= 8 */
n = (int32_t)z;
z -= (double)n;
ih = 0;
if (q0 > 0)
{ /* need iq[jz-1] to determine n */
i = (iq[jz - 1] >> (24 - q0));
n += i;
iq[jz - 1] -= i << (24 - q0);
ih = iq[jz - 1] >> (23 - q0);
}
else if (q0 == 0)
ih = iq[jz - 1] >> 23;
else if (z >= 0.5)
ih = 2;
if (ih > 0)
{ /* q > 0.5 */
n += 1;
carry = 0;
for (i = 0; i < jz; i++)
{ /* compute 1-q */
j = iq[i];
if (carry == 0)
{
if (j != 0)
{
carry = 1;
iq[i] = 0x1000000 - j;
}
}
else
iq[i] = 0xffffff - j;
}
if (q0 > 0)
{ /* rare case: chance is 1 in 12 */
switch (q0)
{
case 1:
iq[jz - 1] &= 0x7fffff;
break;
case 2:
iq[jz - 1] &= 0x3fffff;
break;
}
}
if (ih == 2)
{
z = one - z;
if (carry != 0)
z -= std::scalbn(one, q0);
}
}
/* check if recomputation is needed */
if (z == zero)
{
j = 0;
for (i = jz - 1; i >= jk; i--)
j |= iq[i];
if (j == 0)
{ /* need recomputation */
for (k = 1; iq[jk - k] == 0; k++)
; /* k = no. of terms needed */
for (i = jz + 1; i <= jz + k; i++)
{ /* add q[jz+1] to q[jz+k] */
f[jx + i] = (double)ipio2[jv + i];
for (j = 0, fw = 0.0; j <= jx; j++)
fw += x[j] * f[jx + i - j];
q[i] = fw;
}
jz += k;
goto recompute;
}
}
/* chop off zero terms */
if (z == 0.0)
{
jz -= 1;
q0 -= 24;
while (iq[jz] == 0)
{
jz--;
q0 -= 24;
}
}
else
{ /* break z into 24-bit if necessary */
z = std::scalbn(z, -q0);
if (z >= two24)
{
fw = (double)((int32_t)(twon24 * z));
iq[jz] = (int32_t)(z - two24 * fw);
jz += 1;
q0 += 24;
iq[jz] = (int32_t)fw;
}
else
iq[jz] = (int32_t)z;
}
/* convert integer "bit" chunk to floating-point value */
fw = scalbn(one, q0);
for (i = jz; i >= 0; i--)
{
q[i] = fw * (double)iq[i];
fw *= twon24;
}
/* compute PIo2[0,...,jp]*q[jz,...,0] */
for (i = jz; i >= 0; i--)
{
for (fw = 0.0, k = 0; k <= jp && k <= jz - i; k++)
fw += PIo2[k] * q[i + k];
fq[jz - i] = fw;
}
/* compress fq[] into y[] */
switch (prec)
{
case 0:
fw = 0.0;
for (i = jz; i >= 0; i--)
fw += fq[i];
y[0] = (ih == 0) ? fw : -fw;
break;
case 1:
case 2:
fw = 0.0;
for (i = jz; i >= 0; i--)
fw += fq[i];
y[0] = (ih == 0) ? fw : -fw;
fw = fq[0] - fw;
for (i = 1; i <= jz; i++)
fw += fq[i];
y[1] = (ih == 0) ? fw : -fw;
break;
case 3: /* painful */
for (i = jz; i > 0; i--)
{
fw = fq[i - 1] + fq[i];
fq[i] += fq[i - 1] - fw;
fq[i - 1] = fw;
}
for (i = jz; i > 1; i--)
{
fw = fq[i - 1] + fq[i];
fq[i] += fq[i - 1] - fw;
fq[i - 1] = fw;
}
for (fw = 0.0, i = jz; i >= 2; i--)
fw += fq[i];
if (ih == 0)
{
y[0] = fq[0];
y[1] = fq[1];
y[2] = fw;
}
else
{
y[0] = -fq[0];
y[1] = -fq[1];
y[2] = -fw;
}
}
return n & 7;
}
inline std::int32_t __ieee754_rem_pio2(double x, double* y)
{
static const std::int32_t two_over_pi[] = {
0xA2F983, 0x6E4E44, 0x1529FC, 0x2757D1, 0xF534DD, 0xC0DB62,
0x95993C, 0x439041, 0xFE5163, 0xABDEBB, 0xC561B7, 0x246E3A,
0x424DD2, 0xE00649, 0x2EEA09, 0xD1921C, 0xFE1DEB, 0x1CB129,
0xA73EE8, 0x8235F5, 0x2EBB44, 0x84E99C, 0x7026B4, 0x5F7E41,
0x3991D6, 0x398353, 0x39F49C, 0x845F8B, 0xBDF928, 0x3B1FF8,
0x97FFDE, 0x05980F, 0xEF2F11, 0x8B5A0A, 0x6D1F6D, 0x367ECF,
0x27CB09, 0xB74F46, 0x3F669E, 0x5FEA2D, 0x7527BA, 0xC7EBE5,
0xF17B3D, 0x0739F7, 0x8A5292, 0xEA6BFB, 0x5FB11F, 0x8D5D08,
0x560330, 0x46FC7B, 0x6BABF0, 0xCFBC20, 0x9AF436, 0x1DA9E3,
0x91615E, 0xE61B08, 0x659985, 0x5F14A0, 0x68408D, 0xFFD880,
0x4D7327, 0x310606, 0x1556CA, 0x73A8C9, 0x60E27B, 0xC08C6B,
};
static const std::int32_t npio2_hw[] = {
0x3FF921FB, 0x400921FB, 0x4012D97C, 0x401921FB, 0x401F6A7A, 0x4022D97C,
0x4025FDBB, 0x402921FB, 0x402C463A, 0x402F6A7A, 0x4031475C, 0x4032D97C,
0x40346B9C, 0x4035FDBB, 0x40378FDB, 0x403921FB, 0x403AB41B, 0x403C463A,
0x403DD85A, 0x403F6A7A, 0x40407E4C, 0x4041475C, 0x4042106C, 0x4042D97C,
0x4043A28C, 0x40446B9C, 0x404534AC, 0x4045FDBB, 0x4046C6CB, 0x40478FDB,
0x404858EB, 0x404921FB,
};
/*
* invpio2: 53 bits of 2/pi
* pio2_1: first 33 bit of pi/2
* pio2_1t: pi/2 - pio2_1
* pio2_2: second 33 bit of pi/2
* pio2_2t: pi/2 - (pio2_1+pio2_2)
* pio2_3: third 33 bit of pi/2
* pio2_3t: pi/2 - (pio2_1+pio2_2+pio2_3)
*/
static const double
zero = 0.00000000000000000000e+00, /* 0x00000000, 0x00000000 */
half = 5.00000000000000000000e-01, /* 0x3FE00000, 0x00000000 */
two24 = 1.67772160000000000000e+07, /* 0x41700000, 0x00000000 */
invpio2 = 6.36619772367581382433e-01, /* 0x3FE45F30, 0x6DC9C883 */
pio2_1 = 1.57079632673412561417e+00, /* 0x3FF921FB, 0x54400000 */
pio2_1t = 6.07710050650619224932e-11, /* 0x3DD0B461, 0x1A626331 */
pio2_2 = 6.07710050630396597660e-11, /* 0x3DD0B461, 0x1A600000 */
pio2_2t = 2.02226624879595063154e-21, /* 0x3BA3198A, 0x2E037073 */
pio2_3 = 2.02226624871116645580e-21, /* 0x3BA3198A, 0x2E000000 */
pio2_3t = 8.47842766036889956997e-32; /* 0x397B839A, 0x252049C1 */
double z = 0., w, t, r, fn;
double tx[3];
std::int32_t e0, i, j, nx, n, ix, hx;
std::uint32_t low;
GET_HIGH_WORD(hx, x); /* high word of x */
ix = hx & 0x7fffffff;
if (ix <= 0x3fe921fb) /* |x| ~<= pi/4 , no need for reduction */
{
y[0] = x;
y[1] = 0;
return 0;
}
if (ix < 0x4002d97c)
{ /* |x| < 3pi/4, special case with n=+-1 */
if (hx > 0)
{
z = x - pio2_1;
if (ix != 0x3ff921fb)
{ /* 33+53 bit pi is good enough */
y[0] = z - pio2_1t;
y[1] = (z - y[0]) - pio2_1t;
}
else
{ /* near pi/2, use 33+33+53 bit pi */
z -= pio2_2;
y[0] = z - pio2_2t;
y[1] = (z - y[0]) - pio2_2t;
}
return 1;
}
else
{ /* negative x */
z = x + pio2_1;
if (ix != 0x3ff921fb)
{ /* 33+53 bit pi is good enough */
y[0] = z + pio2_1t;
y[1] = (z - y[0]) + pio2_1t;
}
else
{ /* near pi/2, use 33+33+53 bit pi */
z += pio2_2;
y[0] = z + pio2_2t;
y[1] = (z - y[0]) + pio2_2t;
}
return -1;
}
}
if (ix <= 0x413921fb)
{ /* |x| ~<= 2^19*(pi/2), medium_ size */
t = std::fabs(x);
n = (std::int32_t)(t * invpio2 + half);
fn = (double)n;
r = t - fn * pio2_1;
w = fn * pio2_1t; /* 1st round good to 85 bit */
if ((n < 32) && (n > 0) && (ix != npio2_hw[n - 1]))
{
y[0] = r - w; /* quick check no cancellation */
}
else
{
std::uint32_t high;
j = ix >> 20;
y[0] = r - w;
GET_HIGH_WORD(high, y[0]);
i = j - ((high >> 20) & 0x7ff);
if (i > 16)
{ /* 2nd iteration needed, good to 118 */
t = r;
w = fn * pio2_2;
r = t - w;
w = fn * pio2_2t - ((t - r) - w);
y[0] = r - w;
GET_HIGH_WORD(high, y[0]);
i = j - ((high >> 20) & 0x7ff);
if (i > 49)
{ /* 3rd iteration need, 151 bits acc */
t = r; /* will cover all possible cases */
w = fn * pio2_3;
r = t - w;
w = fn * pio2_3t - ((t - r) - w);
y[0] = r - w;
}
}
}
y[1] = (r - y[0]) - w;
if (hx < 0)
{
y[0] = -y[0];
y[1] = -y[1];
return -n;
}
else
return n;
}
/*
* all other (large) arguments
*/
if (ix >= 0x7ff00000)
{ /* x is inf or NaN */
y[0] = y[1] = x - x;
return 0;
}
/* set z = scalbn(|x|,ilogb(x)-23) */
GET_LOW_WORD(low, x);
SET_LOW_WORD(z, low);
e0 = (ix >> 20) - 1046; /* e0 = ilogb(z)-23; */
SET_HIGH_WORD(z, ix - ((std::int32_t)(e0 << 20)));
for (i = 0; i < 2; i++)
{
tx[i] = (double)((std::int32_t)(z));
z = (z - tx[i]) * two24;
}
tx[2] = z;
nx = 3;
while (tx[nx - 1] == zero)
nx--; /* skip zero term */
n = __kernel_rem_pio2(tx, y, e0, nx, 2, two_over_pi);
if (hx < 0)
{
y[0] = -y[0];
y[1] = -y[1];
return -n;
}
return n;
}
}
#undef XSIMD_LITTLE_ENDIAN
#undef SET_LOW_WORD
#undef SET_HIGH_WORD
#undef GET_LOW_WORD
#undef GET_HIGH_WORD
#undef HIGH_WORD_IDX
#undef LOW_WORD_IDX
#undef ONCE0
}
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