/*
* Copyright (c) 2014 Advanced Micro Devices, Inc.
*
* Permission is hereby granted, free of charge, to any person obtaining a copy
* of this software and associated documentation files (the "Software"), to deal
* in the Software without restriction, including without limitation the rights
* to use, copy, modify, merge, publish, distribute, sublicense, and/or sell
* copies of the Software, and to permit persons to whom the Software is
* furnished to do so, subject to the following conditions:
*
* The above copyright notice and this permission notice shall be included in
* all copies or substantial portions of the Software.
*
* THE SOFTWARE IS PROVIDED "AS IS", WITHOUT WARRANTY OF ANY KIND, EXPRESS OR
* IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES OF MERCHANTABILITY,
* FITNESS FOR A PARTICULAR PURPOSE AND NONINFRINGEMENT. IN NO EVENT SHALL THE
* AUTHORS OR COPYRIGHT HOLDERS BE LIABLE FOR ANY CLAIM, DAMAGES OR OTHER
* LIABILITY, WHETHER IN AN ACTION OF CONTRACT, TORT OR OTHERWISE, ARISING FROM,
* OUT OF OR IN CONNECTION WITH THE SOFTWARE OR THE USE OR OTHER DEALINGS IN
* THE SOFTWARE.
*/
#include <clc/clc.h>
#include "math.h"
#include "tables.h"
#include "sincos_helpers.h"
#define bitalign(hi, lo, shift) \
((hi) << (32 - (shift))) | ((lo) >> (shift));
#define bytealign(src0, src1, src2) \
((uint) (((((long)(src0)) << 32) | (long)(src1)) >> (((src2) & 3)*8)))
_CLC_DEF float __clc_sinf_piby4(float x, float y) {
// Taylor series for sin(x) is x - x^3/3! + x^5/5! - x^7/7! ...
// = x * (1 - x^2/3! + x^4/5! - x^6/7! ...
// = x * f(w)
// where w = x*x and f(w) = (1 - w/3! + w^2/5! - w^3/7! ...
// We use a minimax approximation of (f(w) - 1) / w
// because this produces an expansion in even powers of x.
const float c1 = -0.1666666666e0f;
const float c2 = 0.8333331876e-2f;
const float c3 = -0.198400874e-3f;
const float c4 = 0.272500015e-5f;
const float c5 = -2.5050759689e-08f; // 0xb2d72f34
const float c6 = 1.5896910177e-10f; // 0x2f2ec9d3
float z = x * x;
float v = z * x;
float r = mad(z, mad(z, mad(z, mad(z, c6, c5), c4), c3), c2);
float ret = x - mad(v, -c1, mad(z, mad(y, 0.5f, -v*r), -y));
return ret;
}
_CLC_DEF float __clc_cosf_piby4(float x, float y) {
// Taylor series for cos(x) is 1 - x^2/2! + x^4/4! - x^6/6! ...
// = f(w)
// where w = x*x and f(w) = (1 - w/2! + w^2/4! - w^3/6! ...
// We use a minimax approximation of (f(w) - 1 + w/2) / (w*w)
// because this produces an expansion in even powers of x.
const float c1 = 0.416666666e-1f;
const float c2 = -0.138888876e-2f;
const float c3 = 0.248006008e-4f;
const float c4 = -0.2730101334e-6f;
const float c5 = 2.0875723372e-09f; // 0x310f74f6
const float c6 = -1.1359647598e-11f; // 0xad47d74e
float z = x * x;
float r = z * mad(z, mad(z, mad(z, mad(z, mad(z, c6, c5), c4), c3), c2), c1);
// if |x| < 0.3
float qx = 0.0f;
int ix = as_int(x) & EXSIGNBIT_SP32;
// 0.78125 > |x| >= 0.3
float xby4 = as_float(ix - 0x01000000);
qx = (ix >= 0x3e99999a) & (ix <= 0x3f480000) ? xby4 : qx;
// x > 0.78125
qx = ix > 0x3f480000 ? 0.28125f : qx;
float hz = mad(z, 0.5f, -qx);
float a = 1.0f - qx;
float ret = a - (hz - mad(z, r, -x*y));
return ret;
}
_CLC_DEF float __clc_tanf_piby4(float x, int regn)
{
// Core Remez [1,2] approximation to tan(x) on the interval [0,pi/4].
float r = x * x;
float a = mad(r, -0.0172032480471481694693109f, 0.385296071263995406715129f);
float b = mad(r,
mad(r, 0.01844239256901656082986661f, -0.51396505478854532132342f),
1.15588821434688393452299f);
float t = mad(x*r, native_divide(a, b), x);
float tr = -MATH_RECIP(t);
return regn & 1 ? tr : t;
}
_CLC_DEF void __clc_fullMulS(float *hi, float *lo, float a, float b, float bh, float bt)
{
if (HAVE_HW_FMA32()) {
float ph = a * b;
*hi = ph;
*lo = fma(a, b, -ph);
} else {
float ah = as_float(as_uint(a) & 0xfffff000U);
float at = a - ah;
float ph = a * b;
float pt = mad(at, bt, mad(at, bh, mad(ah, bt, mad(ah, bh, -ph))));
*hi = ph;
*lo = pt;
}
}
_CLC_DEF float __clc_removePi2S(float *hi, float *lo, float x)
{
// 72 bits of pi/2
const float fpiby2_1 = (float) 0xC90FDA / 0x1.0p+23f;
const float fpiby2_1_h = (float) 0xC90 / 0x1.0p+11f;
const float fpiby2_1_t = (float) 0xFDA / 0x1.0p+23f;
const float fpiby2_2 = (float) 0xA22168 / 0x1.0p+47f;
const float fpiby2_2_h = (float) 0xA22 / 0x1.0p+35f;
const float fpiby2_2_t = (float) 0x168 / 0x1.0p+47f;
const float fpiby2_3 = (float) 0xC234C4 / 0x1.0p+71f;
const float fpiby2_3_h = (float) 0xC23 / 0x1.0p+59f;
const float fpiby2_3_t = (float) 0x4C4 / 0x1.0p+71f;
const float twobypi = 0x1.45f306p-1f;
float fnpi2 = trunc(mad(x, twobypi, 0.5f));
// subtract n * pi/2 from x
float rhead, rtail;
__clc_fullMulS(&rhead, &rtail, fnpi2, fpiby2_1, fpiby2_1_h, fpiby2_1_t);
float v = x - rhead;
float rem = v + (((x - v) - rhead) - rtail);
float rhead2, rtail2;
__clc_fullMulS(&rhead2, &rtail2, fnpi2, fpiby2_2, fpiby2_2_h, fpiby2_2_t);
v = rem - rhead2;
rem = v + (((rem - v) - rhead2) - rtail2);
float rhead3, rtail3;
__clc_fullMulS(&rhead3, &rtail3, fnpi2, fpiby2_3, fpiby2_3_h, fpiby2_3_t);
v = rem - rhead3;
*hi = v + ((rem - v) - rhead3);
*lo = -rtail3;
return fnpi2;
}
_CLC_DEF int __clc_argReductionSmallS(float *r, float *rr, float x)
{
float fnpi2 = __clc_removePi2S(r, rr, x);
return (int)fnpi2 & 0x3;
}
#define FULL_MUL(A, B, HI, LO) \
LO = A * B; \
HI = mul_hi(A, B)
#define FULL_MAD(A, B, C, HI, LO) \
LO = ((A) * (B) + (C)); \
HI = mul_hi(A, B); \
HI += LO < C
_CLC_DEF int __clc_argReductionLargeS(float *r, float *rr, float x)
{
int xe = (int)(as_uint(x) >> 23) - 127;
uint xm = 0x00800000U | (as_uint(x) & 0x7fffffU);
// 224 bits of 2/PI: . A2F9836E 4E441529 FC2757D1 F534DDC0 DB629599 3C439041 FE5163AB
const uint b6 = 0xA2F9836EU;
const uint b5 = 0x4E441529U;
const uint b4 = 0xFC2757D1U;
const uint b3 = 0xF534DDC0U;
const uint b2 = 0xDB629599U;
const uint b1 = 0x3C439041U;
const uint b0 = 0xFE5163ABU;
uint p0, p1, p2, p3, p4, p5, p6, p7, c0, c1;
FULL_MUL(xm, b0, c0, p0);
FULL_MAD(xm, b1, c0, c1, p1);
FULL_MAD(xm, b2, c1, c0, p2);
FULL_MAD(xm, b3, c0, c1, p3);
FULL_MAD(xm, b4, c1, c0, p4);
FULL_MAD(xm, b5, c0, c1, p5);
FULL_MAD(xm, b6, c1, p7, p6);
uint fbits = 224 + 23 - xe;
// shift amount to get 2 lsb of integer part at top 2 bits
// min: 25 (xe=18) max: 134 (xe=127)
uint shift = 256U - 2 - fbits;
// Shift by up to 134/32 = 4 words
int c = shift > 31;
p7 = c ? p6 : p7;
p6 = c ? p5 : p6;
p5 = c ? p4 : p5;
p4 = c ? p3 : p4;
p3 = c ? p2 : p3;
p2 = c ? p1 : p2;
p1 = c ? p0 : p1;
shift -= (-c) & 32;
c = shift > 31;
p7 = c ? p6 : p7;
p6 = c ? p5 : p6;
p5 = c ? p4 : p5;
p4 = c ? p3 : p4;
p3 = c ? p2 : p3;
p2 = c ? p1 : p2;
shift -= (-c) & 32;
c = shift > 31;
p7 = c ? p6 : p7;
p6 = c ? p5 : p6;
p5 = c ? p4 : p5;
p4 = c ? p3 : p4;
p3 = c ? p2 : p3;
shift -= (-c) & 32;
c = shift > 31;
p7 = c ? p6 : p7;
p6 = c ? p5 : p6;
p5 = c ? p4 : p5;
p4 = c ? p3 : p4;
shift -= (-c) & 32;
// bitalign cannot handle a shift of 32
c = shift > 0;
shift = 32 - shift;
uint t7 = bitalign(p7, p6, shift);
uint t6 = bitalign(p6, p5, shift);
uint t5 = bitalign(p5, p4, shift);
p7 = c ? t7 : p7;
p6 = c ? t6 : p6;
p5 = c ? t5 : p5;
// Get 2 lsb of int part and msb of fraction
int i = p7 >> 29;
// Scoot up 2 more bits so only fraction remains
p7 = bitalign(p7, p6, 30);
p6 = bitalign(p6, p5, 30);
p5 = bitalign(p5, p4, 30);
// Subtract 1 if msb of fraction is 1, i.e. fraction >= 0.5
uint flip = i & 1 ? 0xffffffffU : 0U;
uint sign = i & 1 ? 0x80000000U : 0U;
p7 = p7 ^ flip;
p6 = p6 ^ flip;
p5 = p5 ^ flip;
// Find exponent and shift away leading zeroes and hidden bit
xe = clz(p7) + 1;
shift = 32 - xe;
p7 = bitalign(p7, p6, shift);
p6 = bitalign(p6, p5, shift);
// Most significant part of fraction
float q1 = as_float(sign | ((127 - xe) << 23) | (p7 >> 9));
// Shift out bits we captured on q1
p7 = bitalign(p7, p6, 32-23);
// Get 24 more bits of fraction in another float, there are not long strings of zeroes here
int xxe = clz(p7) + 1;
p7 = bitalign(p7, p6, 32-xxe);
float q0 = as_float(sign | ((127 - (xe + 23 + xxe)) << 23) | (p7 >> 9));
// At this point, the fraction q1 + q0 is correct to at least 48 bits
// Now we need to multiply the fraction by pi/2
// This loses us about 4 bits
// pi/2 = C90 FDA A22 168 C23 4C4
const float pio2h = (float)0xc90fda / 0x1.0p+23f;
const float pio2hh = (float)0xc90 / 0x1.0p+11f;
const float pio2ht = (float)0xfda / 0x1.0p+23f;
const float pio2t = (float)0xa22168 / 0x1.0p+47f;
float rh, rt;
if (HAVE_HW_FMA32()) {
rh = q1 * pio2h;
rt = fma(q0, pio2h, fma(q1, pio2t, fma(q1, pio2h, -rh)));
} else {
float q1h = as_float(as_uint(q1) & 0xfffff000);
float q1t = q1 - q1h;
rh = q1 * pio2h;
rt = mad(q1t, pio2ht, mad(q1t, pio2hh, mad(q1h, pio2ht, mad(q1h, pio2hh, -rh))));
rt = mad(q0, pio2h, mad(q1, pio2t, rt));
}
float t = rh + rt;
rt = rt - (t - rh);
*r = t;
*rr = rt;
return ((i >> 1) + (i & 1)) & 0x3;
}
_CLC_DEF int __clc_argReductionS(float *r, float *rr, float x)
{
if (x < 0x1.0p+23f)
return __clc_argReductionSmallS(r, rr, x);
else
return __clc_argReductionLargeS(r, rr, x);
}
#ifdef cl_khr_fp64
#pragma OPENCL EXTENSION cl_khr_fp64 : enable
// Reduction for medium sized arguments
_CLC_DEF void __clc_remainder_piby2_medium(double x, double *r, double *rr, int *regn) {
// How many pi/2 is x a multiple of?
const double two_by_pi = 0x1.45f306dc9c883p-1;
double dnpi2 = trunc(fma(x, two_by_pi, 0.5));
const double piby2_h = -7074237752028440.0 / 0x1.0p+52;
const double piby2_m = -2483878800010755.0 / 0x1.0p+105;
const double piby2_t = -3956492004828932.0 / 0x1.0p+158;
// Compute product of npi2 with 159 bits of 2/pi
double p_hh = piby2_h * dnpi2;
double p_ht = fma(piby2_h, dnpi2, -p_hh);
double p_mh = piby2_m * dnpi2;
double p_mt = fma(piby2_m, dnpi2, -p_mh);
double p_th = piby2_t * dnpi2;
double p_tt = fma(piby2_t, dnpi2, -p_th);
// Reduce to 159 bits
double ph = p_hh;
double pm = p_ht + p_mh;
double t = p_mh - (pm - p_ht);
double pt = p_th + t + p_mt + p_tt;
t = ph + pm; pm = pm - (t - ph); ph = t;
t = pm + pt; pt = pt - (t - pm); pm = t;
// Subtract from x
t = x + ph;
double qh = t + pm;
double qt = pm - (qh - t) + pt;
*r = qh;
*rr = qt;
*regn = (int)(long)dnpi2 & 0x3;
}
// Given positive argument x, reduce it to the range [-pi/4,pi/4] using
// extra precision, and return the result in r, rr.
// Return value "regn" tells how many lots of pi/2 were subtracted
// from x to put it in the range [-pi/4,pi/4], mod 4.
_CLC_DEF void __clc_remainder_piby2_large(double x, double *r, double *rr, int *regn) {
long ux = as_long(x);
int e = (int)(ux >> 52) - 1023;
int i = max(23, (e >> 3) + 17);
int j = 150 - i;
int j16 = j & ~0xf;
double fract_temp;
// The following extracts 192 consecutive bits of 2/pi aligned on an arbitrary byte boundary
uint4 q0 = USE_TABLE(pibits_tbl, j16);
uint4 q1 = USE_TABLE(pibits_tbl, (j16 + 16));
uint4 q2 = USE_TABLE(pibits_tbl, (j16 + 32));
int k = (j >> 2) & 0x3;
int4 c = (int4)k == (int4)(0, 1, 2, 3);
uint u0, u1, u2, u3, u4, u5, u6;
u0 = c.s1 ? q0.s1 : q0.s0;
u0 = c.s2 ? q0.s2 : u0;
u0 = c.s3 ? q0.s3 : u0;
u1 = c.s1 ? q0.s2 : q0.s1;
u1 = c.s2 ? q0.s3 : u1;
u1 = c.s3 ? q1.s0 : u1;
u2 = c.s1 ? q0.s3 : q0.s2;
u2 = c.s2 ? q1.s0 : u2;
u2 = c.s3 ? q1.s1 : u2;
u3 = c.s1 ? q1.s0 : q0.s3;
u3 = c.s2 ? q1.s1 : u3;
u3 = c.s3 ? q1.s2 : u3;
u4 = c.s1 ? q1.s1 : q1.s0;
u4 = c.s2 ? q1.s2 : u4;
u4 = c.s3 ? q1.s3 : u4;
u5 = c.s1 ? q1.s2 : q1.s1;
u5 = c.s2 ? q1.s3 : u5;
u5 = c.s3 ? q2.s0 : u5;
u6 = c.s1 ? q1.s3 : q1.s2;
u6 = c.s2 ? q2.s0 : u6;
u6 = c.s3 ? q2.s1 : u6;
uint v0 = bytealign(u1, u0, j);
uint v1 = bytealign(u2, u1, j);
uint v2 = bytealign(u3, u2, j);
uint v3 = bytealign(u4, u3, j);
uint v4 = bytealign(u5, u4, j);
uint v5 = bytealign(u6, u5, j);
// Place those 192 bits in 4 48-bit doubles along with correct exponent
// If i > 1018 we would get subnormals so we scale p up and x down to get the same product
i = 2 + 8*i;
x *= i > 1018 ? 0x1.0p-136 : 1.0;
i -= i > 1018 ? 136 : 0;
uint ua = (uint)(1023 + 52 - i) << 20;
double a = as_double((uint2)(0, ua));
double p0 = as_double((uint2)(v0, ua | (v1 & 0xffffU))) - a;
ua += 0x03000000U;
a = as_double((uint2)(0, ua));
double p1 = as_double((uint2)((v2 << 16) | (v1 >> 16), ua | (v2 >> 16))) - a;
ua += 0x03000000U;
a = as_double((uint2)(0, ua));
double p2 = as_double((uint2)(v3, ua | (v4 & 0xffffU))) - a;
ua += 0x03000000U;
a = as_double((uint2)(0, ua));
double p3 = as_double((uint2)((v5 << 16) | (v4 >> 16), ua | (v5 >> 16))) - a;
// Exact multiply
double f0h = p0 * x;
double f0l = fma(p0, x, -f0h);
double f1h = p1 * x;
double f1l = fma(p1, x, -f1h);
double f2h = p2 * x;
double f2l = fma(p2, x, -f2h);
double f3h = p3 * x;
double f3l = fma(p3, x, -f3h);
// Accumulate product into 4 doubles
double s, t;
double f3 = f3h + f2h;
t = f2h - (f3 - f3h);
s = f3l + t;
t = t - (s - f3l);
double f2 = s + f1h;
t = f1h - (f2 - s) + t;
s = f2l + t;
t = t - (s - f2l);
double f1 = s + f0h;
t = f0h - (f1 - s) + t;
s = f1l + t;
double f0 = s + f0l;
// Strip off unwanted large integer bits
f3 = 0x1.0p+10 * fract(f3 * 0x1.0p-10, &fract_temp);
f3 += f3 + f2 < 0.0 ? 0x1.0p+10 : 0.0;
// Compute least significant integer bits
t = f3 + f2;
double di = t - fract(t, &fract_temp);
i = (float)di;
// Shift out remaining integer part
f3 -= di;
s = f3 + f2; t = f2 - (s - f3); f3 = s; f2 = t;
s = f2 + f1; t = f1 - (s - f2); f2 = s; f1 = t;
f1 += f0;
// Subtract 1 if fraction is >= 0.5, and update regn
int g = f3 >= 0.5;
i += g;
f3 -= (float)g;
// Shift up bits
s = f3 + f2; t = f2 -(s - f3); f3 = s; f2 = t + f1;
// Multiply precise fraction by pi/2 to get radians
const double p2h = 7074237752028440.0 / 0x1.0p+52;
const double p2t = 4967757600021510.0 / 0x1.0p+106;
double rhi = f3 * p2h;
double rlo = fma(f2, p2h, fma(f3, p2t, fma(f3, p2h, -rhi)));
*r = rhi + rlo;
*rr = rlo - (*r - rhi);
*regn = i & 0x3;
}
_CLC_DEF double2 __clc_sincos_piby4(double x, double xx) {
// Taylor series for sin(x) is x - x^3/3! + x^5/5! - x^7/7! ...
// = x * (1 - x^2/3! + x^4/5! - x^6/7! ...
// = x * f(w)
// where w = x*x and f(w) = (1 - w/3! + w^2/5! - w^3/7! ...
// We use a minimax approximation of (f(w) - 1) / w
// because this produces an expansion in even powers of x.
// If xx (the tail of x) is non-zero, we add a correction
// term g(x,xx) = (1-x*x/2)*xx to the result, where g(x,xx)
// is an approximation to cos(x)*sin(xx) valid because
// xx is tiny relative to x.
// Taylor series for cos(x) is 1 - x^2/2! + x^4/4! - x^6/6! ...
// = f(w)
// where w = x*x and f(w) = (1 - w/2! + w^2/4! - w^3/6! ...
// We use a minimax approximation of (f(w) - 1 + w/2) / (w*w)
// because this produces an expansion in even powers of x.
// If xx (the tail of x) is non-zero, we subtract a correction
// term g(x,xx) = x*xx to the result, where g(x,xx)
// is an approximation to sin(x)*sin(xx) valid because
// xx is tiny relative to x.
const double sc1 = -0.166666666666666646259241729;
const double sc2 = 0.833333333333095043065222816e-2;
const double sc3 = -0.19841269836761125688538679e-3;
const double sc4 = 0.275573161037288022676895908448e-5;
const double sc5 = -0.25051132068021699772257377197e-7;
const double sc6 = 0.159181443044859136852668200e-9;
const double cc1 = 0.41666666666666665390037e-1;
const double cc2 = -0.13888888888887398280412e-2;
const double cc3 = 0.248015872987670414957399e-4;
const double cc4 = -0.275573172723441909470836e-6;
const double cc5 = 0.208761463822329611076335e-8;
const double cc6 = -0.113826398067944859590880e-10;
double x2 = x * x;
double x3 = x2 * x;
double r = 0.5 * x2;
double t = 1.0 - r;
double sp = fma(fma(fma(fma(sc6, x2, sc5), x2, sc4), x2, sc3), x2, sc2);
double cp = t + fma(fma(fma(fma(fma(fma(cc6, x2, cc5), x2, cc4), x2, cc3), x2, cc2), x2, cc1),
x2*x2, fma(x, xx, (1.0 - t) - r));
double2 ret;
ret.lo = x - fma(-x3, sc1, fma(fma(-x3, sp, 0.5*xx), x2, -xx));
ret.hi = cp;
return ret;
}
#endif
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