309 lines
9.0 KiB
Common Lisp
309 lines
9.0 KiB
Common Lisp
/*
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* Copyright (c) 2014 Advanced Micro Devices, Inc.
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*
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* Permission is hereby granted, free of charge, to any person obtaining a copy
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* of this software and associated documentation files (the "Software"), to deal
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* in the Software without restriction, including without limitation the rights
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* to use, copy, modify, merge, publish, distribute, sublicense, and/or sell
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* copies of the Software, and to permit persons to whom the Software is
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* furnished to do so, subject to the following conditions:
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*
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* The above copyright notice and this permission notice shall be included in
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* all copies or substantial portions of the Software.
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*
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* THE SOFTWARE IS PROVIDED "AS IS", WITHOUT WARRANTY OF ANY KIND, EXPRESS OR
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* IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES OF MERCHANTABILITY,
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* FITNESS FOR A PARTICULAR PURPOSE AND NONINFRINGEMENT. IN NO EVENT SHALL THE
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* AUTHORS OR COPYRIGHT HOLDERS BE LIABLE FOR ANY CLAIM, DAMAGES OR OTHER
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* LIABILITY, WHETHER IN AN ACTION OF CONTRACT, TORT OR OTHERWISE, ARISING FROM,
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* OUT OF OR IN CONNECTION WITH THE SOFTWARE OR THE USE OR OTHER DEALINGS IN
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* THE SOFTWARE.
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*/
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#include <clc/clc.h>
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#include "math.h"
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#include "sincos_helpers.h"
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uint bitalign(uint hi, uint lo, uint shift)
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{
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return (hi << (32 - shift)) | (lo >> shift);
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}
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float sinf_piby4(float x, float y)
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{
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// Taylor series for sin(x) is x - x^3/3! + x^5/5! - x^7/7! ...
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// = x * (1 - x^2/3! + x^4/5! - x^6/7! ...
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// = x * f(w)
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// where w = x*x and f(w) = (1 - w/3! + w^2/5! - w^3/7! ...
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// We use a minimax approximation of (f(w) - 1) / w
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// because this produces an expansion in even powers of x.
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const float c1 = -0.1666666666e0f;
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const float c2 = 0.8333331876e-2f;
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const float c3 = -0.198400874e-3f;
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const float c4 = 0.272500015e-5f;
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const float c5 = -2.5050759689e-08f; // 0xb2d72f34
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const float c6 = 1.5896910177e-10f; // 0x2f2ec9d3
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float z = x * x;
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float v = z * x;
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float r = mad(z, mad(z, mad(z, mad(z, c6, c5), c4), c3), c2);
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float ret = x - mad(v, -c1, mad(z, mad(y, 0.5f, -v*r), -y));
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return ret;
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}
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float cosf_piby4(float x, float y)
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{
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// Taylor series for cos(x) is 1 - x^2/2! + x^4/4! - x^6/6! ...
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// = f(w)
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// where w = x*x and f(w) = (1 - w/2! + w^2/4! - w^3/6! ...
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// We use a minimax approximation of (f(w) - 1 + w/2) / (w*w)
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// because this produces an expansion in even powers of x.
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const float c1 = 0.416666666e-1f;
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const float c2 = -0.138888876e-2f;
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const float c3 = 0.248006008e-4f;
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const float c4 = -0.2730101334e-6f;
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const float c5 = 2.0875723372e-09f; // 0x310f74f6
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const float c6 = -1.1359647598e-11f; // 0xad47d74e
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float z = x * x;
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float r = z * mad(z, mad(z, mad(z, mad(z, mad(z, c6, c5), c4), c3), c2), c1);
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// if |x| < 0.3
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float qx = 0.0f;
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int ix = as_int(x) & EXSIGNBIT_SP32;
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// 0.78125 > |x| >= 0.3
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float xby4 = as_float(ix - 0x01000000);
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qx = (ix >= 0x3e99999a) & (ix <= 0x3f480000) ? xby4 : qx;
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// x > 0.78125
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qx = ix > 0x3f480000 ? 0.28125f : qx;
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float hz = mad(z, 0.5f, -qx);
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float a = 1.0f - qx;
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float ret = a - (hz - mad(z, r, -x*y));
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return ret;
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}
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void fullMulS(float *hi, float *lo, float a, float b, float bh, float bt)
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{
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if (HAVE_HW_FMA32()) {
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float ph = a * b;
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*hi = ph;
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*lo = fma(a, b, -ph);
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} else {
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float ah = as_float(as_uint(a) & 0xfffff000U);
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float at = a - ah;
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float ph = a * b;
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float pt = mad(at, bt, mad(at, bh, mad(ah, bt, mad(ah, bh, -ph))));
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*hi = ph;
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*lo = pt;
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}
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}
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float removePi2S(float *hi, float *lo, float x)
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{
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// 72 bits of pi/2
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const float fpiby2_1 = (float) 0xC90FDA / 0x1.0p+23f;
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const float fpiby2_1_h = (float) 0xC90 / 0x1.0p+11f;
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const float fpiby2_1_t = (float) 0xFDA / 0x1.0p+23f;
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const float fpiby2_2 = (float) 0xA22168 / 0x1.0p+47f;
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const float fpiby2_2_h = (float) 0xA22 / 0x1.0p+35f;
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const float fpiby2_2_t = (float) 0x168 / 0x1.0p+47f;
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const float fpiby2_3 = (float) 0xC234C4 / 0x1.0p+71f;
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const float fpiby2_3_h = (float) 0xC23 / 0x1.0p+59f;
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const float fpiby2_3_t = (float) 0x4C4 / 0x1.0p+71f;
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const float twobypi = 0x1.45f306p-1f;
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float fnpi2 = trunc(mad(x, twobypi, 0.5f));
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// subtract n * pi/2 from x
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float rhead, rtail;
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fullMulS(&rhead, &rtail, fnpi2, fpiby2_1, fpiby2_1_h, fpiby2_1_t);
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float v = x - rhead;
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float rem = v + (((x - v) - rhead) - rtail);
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float rhead2, rtail2;
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fullMulS(&rhead2, &rtail2, fnpi2, fpiby2_2, fpiby2_2_h, fpiby2_2_t);
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v = rem - rhead2;
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rem = v + (((rem - v) - rhead2) - rtail2);
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float rhead3, rtail3;
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fullMulS(&rhead3, &rtail3, fnpi2, fpiby2_3, fpiby2_3_h, fpiby2_3_t);
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v = rem - rhead3;
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*hi = v + ((rem - v) - rhead3);
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*lo = -rtail3;
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return fnpi2;
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}
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int argReductionSmallS(float *r, float *rr, float x)
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{
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float fnpi2 = removePi2S(r, rr, x);
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return (int)fnpi2 & 0x3;
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}
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#define FULL_MUL(A, B, HI, LO) \
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LO = A * B; \
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HI = mul_hi(A, B)
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#define FULL_MAD(A, B, C, HI, LO) \
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LO = ((A) * (B) + (C)); \
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HI = mul_hi(A, B); \
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HI += LO < C
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int argReductionLargeS(float *r, float *rr, float x)
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{
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int xe = (int)(as_uint(x) >> 23) - 127;
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uint xm = 0x00800000U | (as_uint(x) & 0x7fffffU);
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// 224 bits of 2/PI: . A2F9836E 4E441529 FC2757D1 F534DDC0 DB629599 3C439041 FE5163AB
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const uint b6 = 0xA2F9836EU;
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const uint b5 = 0x4E441529U;
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const uint b4 = 0xFC2757D1U;
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const uint b3 = 0xF534DDC0U;
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const uint b2 = 0xDB629599U;
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const uint b1 = 0x3C439041U;
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const uint b0 = 0xFE5163ABU;
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uint p0, p1, p2, p3, p4, p5, p6, p7, c0, c1;
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FULL_MUL(xm, b0, c0, p0);
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FULL_MAD(xm, b1, c0, c1, p1);
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FULL_MAD(xm, b2, c1, c0, p2);
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FULL_MAD(xm, b3, c0, c1, p3);
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FULL_MAD(xm, b4, c1, c0, p4);
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FULL_MAD(xm, b5, c0, c1, p5);
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FULL_MAD(xm, b6, c1, p7, p6);
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uint fbits = 224 + 23 - xe;
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// shift amount to get 2 lsb of integer part at top 2 bits
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// min: 25 (xe=18) max: 134 (xe=127)
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uint shift = 256U - 2 - fbits;
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// Shift by up to 134/32 = 4 words
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int c = shift > 31;
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p7 = c ? p6 : p7;
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p6 = c ? p5 : p6;
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p5 = c ? p4 : p5;
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p4 = c ? p3 : p4;
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p3 = c ? p2 : p3;
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p2 = c ? p1 : p2;
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p1 = c ? p0 : p1;
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shift -= (-c) & 32;
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c = shift > 31;
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p7 = c ? p6 : p7;
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p6 = c ? p5 : p6;
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p5 = c ? p4 : p5;
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p4 = c ? p3 : p4;
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p3 = c ? p2 : p3;
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p2 = c ? p1 : p2;
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shift -= (-c) & 32;
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c = shift > 31;
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p7 = c ? p6 : p7;
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p6 = c ? p5 : p6;
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p5 = c ? p4 : p5;
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p4 = c ? p3 : p4;
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p3 = c ? p2 : p3;
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shift -= (-c) & 32;
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c = shift > 31;
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p7 = c ? p6 : p7;
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p6 = c ? p5 : p6;
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p5 = c ? p4 : p5;
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p4 = c ? p3 : p4;
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shift -= (-c) & 32;
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// bitalign cannot handle a shift of 32
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c = shift > 0;
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shift = 32 - shift;
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uint t7 = bitalign(p7, p6, shift);
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uint t6 = bitalign(p6, p5, shift);
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uint t5 = bitalign(p5, p4, shift);
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p7 = c ? t7 : p7;
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p6 = c ? t6 : p6;
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p5 = c ? t5 : p5;
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// Get 2 lsb of int part and msb of fraction
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int i = p7 >> 29;
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// Scoot up 2 more bits so only fraction remains
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p7 = bitalign(p7, p6, 30);
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p6 = bitalign(p6, p5, 30);
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p5 = bitalign(p5, p4, 30);
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// Subtract 1 if msb of fraction is 1, i.e. fraction >= 0.5
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uint flip = i & 1 ? 0xffffffffU : 0U;
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uint sign = i & 1 ? 0x80000000U : 0U;
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p7 = p7 ^ flip;
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p6 = p6 ^ flip;
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p5 = p5 ^ flip;
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// Find exponent and shift away leading zeroes and hidden bit
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xe = clz(p7) + 1;
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shift = 32 - xe;
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p7 = bitalign(p7, p6, shift);
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p6 = bitalign(p6, p5, shift);
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// Most significant part of fraction
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float q1 = as_float(sign | ((127 - xe) << 23) | (p7 >> 9));
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// Shift out bits we captured on q1
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p7 = bitalign(p7, p6, 32-23);
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// Get 24 more bits of fraction in another float, there are not long strings of zeroes here
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int xxe = clz(p7) + 1;
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p7 = bitalign(p7, p6, 32-xxe);
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float q0 = as_float(sign | ((127 - (xe + 23 + xxe)) << 23) | (p7 >> 9));
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// At this point, the fraction q1 + q0 is correct to at least 48 bits
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// Now we need to multiply the fraction by pi/2
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// This loses us about 4 bits
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// pi/2 = C90 FDA A22 168 C23 4C4
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const float pio2h = (float)0xc90fda / 0x1.0p+23f;
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const float pio2hh = (float)0xc90 / 0x1.0p+11f;
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const float pio2ht = (float)0xfda / 0x1.0p+23f;
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const float pio2t = (float)0xa22168 / 0x1.0p+47f;
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float rh, rt;
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if (HAVE_HW_FMA32()) {
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rh = q1 * pio2h;
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rt = fma(q0, pio2h, fma(q1, pio2t, fma(q1, pio2h, -rh)));
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} else {
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float q1h = as_float(as_uint(q1) & 0xfffff000);
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float q1t = q1 - q1h;
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rh = q1 * pio2h;
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rt = mad(q1t, pio2ht, mad(q1t, pio2hh, mad(q1h, pio2ht, mad(q1h, pio2hh, -rh))));
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rt = mad(q0, pio2h, mad(q1, pio2t, rt));
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}
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float t = rh + rt;
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rt = rt - (t - rh);
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*r = t;
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*rr = rt;
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return ((i >> 1) + (i & 1)) & 0x3;
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}
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int argReductionS(float *r, float *rr, float x)
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{
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if (x < 0x1.0p+23f)
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return argReductionSmallS(r, rr, x);
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else
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return argReductionLargeS(r, rr, x);
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}
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