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565 lines
14 KiB
ArmAsm
565 lines
14 KiB
ArmAsm
.file "exp2.s"
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// Copyright (c) 2000 - 2005, Intel Corporation
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// All rights reserved.
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//
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// Contributed 2000 by the Intel Numerics Group, Intel Corporation
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//
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// Redistribution and use in source and binary forms, with or without
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// modification, are permitted provided that the following conditions are
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// met:
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//
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// * Redistributions of source code must retain the above copyright
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// notice, this list of conditions and the following disclaimer.
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//
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// * Redistributions in binary form must reproduce the above copyright
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// notice, this list of conditions and the following disclaimer in the
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// documentation and/or other materials provided with the distribution.
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//
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// * The name of Intel Corporation may not be used to endorse or promote
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// products derived from this software without specific prior written
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// permission.
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// THIS SOFTWARE IS PROVIDED BY THE COPYRIGHT HOLDERS AND CONTRIBUTORS
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// "AS IS" AND ANY EXPRESS OR IMPLIED WARRANTIES, INCLUDING, BUT NOT
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// LIMITED TO, THE IMPLIED WARRANTIES OF MERCHANTABILITY AND FITNESS FOR
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// A PARTICULAR PURPOSE ARE DISCLAIMED. IN NO EVENT SHALL INTEL OR ITS
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// CONTRIBUTORS BE LIABLE FOR ANY DIRECT, INDIRECT, INCIDENTAL, SPECIAL,
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// EXEMPLARY, OR CONSEQUENTIAL DAMAGES (INCLUDING, BUT NOT LIMITED TO,
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// PROCUREMENT OF SUBSTITUTE GOODS OR SERVICES; LOSS OF USE, DATA, OR
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// PROFITS; OR BUSINESS INTERRUPTION) HOWEVER CAUSED AND ON ANY THEORY
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// OF LIABILITY, WHETHER IN CONTRACT, STRICT LIABILITY OR TORT (INCLUDING
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// NEGLIGENCE OR OTHERWISE) ARISING IN ANY WAY OUT OF THE USE OF THIS
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// SOFTWARE, EVEN IF ADVISED OF THE POSSIBILITY OF SUCH DAMAGE.
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//
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// Intel Corporation is the author of this code, and requests that all
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// problem reports or change requests be submitted to it directly at
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// http://www.intel.com/software/products/opensource/libraries/num.htm.
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//
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// History
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//==============================================================
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// 08/25/00 Initial version
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// 05/20/02 Cleaned up namespace and sf0 syntax
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// 09/05/02 Improved performance
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// 01/17/03 Fixed to call error support when x=1024.0
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// 03/31/05 Reformatted delimiters between data tables
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//
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// API
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//==============================================================
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// double exp2(double)
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//
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// Overview of operation
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//==============================================================
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// Background
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//
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// Implementation
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//
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// Let x= (K + fh + fl + r), where
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// K is an integer, fh= 0.b1 b2 b3 b4 b5,
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// fl= 2^{-5}* 0.b6 b7 b8 b8 b10 (fh, fl >= 0),
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// and |r|<2^{-11}
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// Th is a table that stores 2^fh (32 entries) rounded to
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// double extended precision (only mantissa is stored)
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// Tl is a table that stores 2^fl (32 entries) rounded to
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// double extended precision (only mantissa is stored)
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//
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// 2^x is approximated as
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// 2^K * Th [ f ] * Tl [ f ] * (1+c1*r+c2*r^2+c3*r^3+c4*r^4)
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// Note: We use the following trick to speed up conversion from FP to integer:
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//
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// Let x = K + r, where K is an integer, and |r| <= 0.5
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// Let N be the number of significand bits for the FP format used
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// ( N=64 for double-extended, N=53 for double)
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//
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// Then let y = 1.5 * 2^(N-1) + x for RN mode
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// K = y - 1.5 * 2^(N-1)
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// r = x - K
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//
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// If we want to obtain the integer part and the first m fractional bits of x,
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// we can use the same trick, but with a constant of 1.5 * 2^(N-1-m):
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//
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// Let x = K + f + r
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// f = 0.b_1 b_2 ... b_m
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// |r| <= 2^(-m-1)
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//
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// Then let y = 1.5 * 2^(N-1-m) + x for RN mode
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// (K+f) = y - 1.5 * 2^(N-1-m)
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// r = x - K
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// Special values
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//==============================================================
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// exp2(0)= 1
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// exp2(+inf)= inf
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// exp2(-inf)= 0
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//
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// Registers used
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//==============================================================
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// r2-r3, r14-r40
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// f6-f15, f32-f45
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// p6-p8, p12
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//
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GR_TBL_START = r2
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GR_LOG_TBL = r3
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GR_OF_LIMIT = r14
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GR_UF_LIMIT = r15
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GR_EXP_CORR = r16
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GR_F_low = r17
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GR_F_high = r18
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GR_K = r19
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GR_Flow_ADDR = r20
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GR_BIAS = r21
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GR_Fh = r22
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GR_Fh_ADDR = r23
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GR_EXPMAX = r24
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GR_EMIN = r25
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GR_ROUNDVAL = r26
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GR_MASK = r27
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GR_KF0 = r28
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GR_MASK_low = r29
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GR_COEFF_START = r30
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GR_SAVE_B0 = r33
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GR_SAVE_PFS = r34
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GR_SAVE_GP = r35
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GR_SAVE_SP = r36
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GR_Parameter_X = r37
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GR_Parameter_Y = r38
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GR_Parameter_RESULT = r39
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GR_Parameter_TAG = r40
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FR_X = f10
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FR_Y = f1
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FR_RESULT = f8
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FR_COEFF1 = f6
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FR_COEFF2 = f7
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FR_R = f9
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FR_KF0 = f12
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FR_COEFF3 = f13
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FR_COEFF4 = f14
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FR_UF_LIMIT = f15
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FR_OF_LIMIT = f32
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FR_EXPMIN = f33
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FR_ROUNDVAL = f34
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FR_KF = f35
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FR_2_TO_K = f36
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FR_T_low = f37
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FR_T_high = f38
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FR_P34 = f39
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FR_R2 = f40
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FR_P12 = f41
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FR_T_low_K = f42
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FR_P14 = f43
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FR_T = f44
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FR_P = f45
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// Data tables
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//==============================================================
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RODATA
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.align 16
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LOCAL_OBJECT_START(poly_coeffs)
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data8 0x3fac6b08d704a0c0, 0x3f83b2ab6fba4e77 // C_3 and C_4
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data8 0xb17217f7d1cf79ab, 0x00003ffe // C_1
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data8 0xf5fdeffc162c7541, 0x00003ffc // C_2
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LOCAL_OBJECT_END(poly_coeffs)
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LOCAL_OBJECT_START(T_table)
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// 2^{0.00000 b6 b7 b8 b9 b10}
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data8 0x8000000000000000, 0x8016302f17467628
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data8 0x802c6436d0e04f50, 0x80429c17d77c18ed
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data8 0x8058d7d2d5e5f6b0, 0x806f17687707a7af
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data8 0x80855ad965e88b83, 0x809ba2264dada76a
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data8 0x80b1ed4fd999ab6c, 0x80c83c56b50cf77f
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data8 0x80de8f3b8b85a0af, 0x80f4e5ff089f763e
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data8 0x810b40a1d81406d4, 0x81219f24a5baa59d
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data8 0x813801881d886f7b, 0x814e67cceb90502c
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data8 0x8164d1f3bc030773, 0x817b3ffd3b2f2e47
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data8 0x8191b1ea15813bfd, 0x81a827baf7838b78
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data8 0x81bea1708dde6055, 0x81d51f0b8557ec1c
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data8 0x81eba08c8ad4536f, 0x820225f44b55b33b
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data8 0x8218af4373fc25eb, 0x822f3c7ab205c89a
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data8 0x8245cd9ab2cec048, 0x825c62a423d13f0c
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data8 0x8272fb97b2a5894c, 0x828998760d01faf3
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data8 0x82a0393fe0bb0ca8, 0x82b6ddf5dbc35906
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//
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// 2^{0.b1 b2 b3 b4 b5}
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data8 0x8000000000000000, 0x82cd8698ac2ba1d7
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data8 0x85aac367cc487b14, 0x88980e8092da8527
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data8 0x8b95c1e3ea8bd6e6, 0x8ea4398b45cd53c0
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data8 0x91c3d373ab11c336, 0x94f4efa8fef70961
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data8 0x9837f0518db8a96f, 0x9b8d39b9d54e5538
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data8 0x9ef5326091a111ad, 0xa27043030c496818
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data8 0xa5fed6a9b15138ea, 0xa9a15ab4ea7c0ef8
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data8 0xad583eea42a14ac6, 0xb123f581d2ac258f
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data8 0xb504f333f9de6484, 0xb8fbaf4762fb9ee9
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data8 0xbd08a39f580c36be, 0xc12c4cca66709456
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data8 0xc5672a115506dadd, 0xc9b9bd866e2f27a2
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data8 0xce248c151f8480e3, 0xd2a81d91f12ae45a
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data8 0xd744fccad69d6af4, 0xdbfbb797daf23755
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data8 0xe0ccdeec2a94e111, 0xe5b906e77c8348a8
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data8 0xeac0c6e7dd24392e, 0xefe4b99bdcdaf5cb
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data8 0xf5257d152486cc2c, 0xfa83b2db722a033a
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LOCAL_OBJECT_END(T_table)
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.section .text
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GLOBAL_LIBM_ENTRY(exp2)
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{.mfi
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alloc r32= ar.pfs, 1, 4, 4, 0
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// will continue only for non-zero normal/denormal numbers
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fclass.nm p12, p0= f8, 0x1b
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// GR_TBL_START= pointer to C_1...C_4 followed by T_table
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addl GR_TBL_START= @ltoff(poly_coeffs), gp
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}
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{.mlx
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mov GR_OF_LIMIT= 0xffff + 10 // Exponent of overflow limit
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movl GR_ROUNDVAL= 0x5a400000 // 1.5*2^(63-10) (SP)
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}
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;;
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// Form special constant 1.5*2^(63-10) to give integer part and first 10
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// fractional bits of x
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{.mfi
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setf.s FR_ROUNDVAL= GR_ROUNDVAL // Form special constant
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fcmp.lt.s1 p6, p8= f8, f0 // X<0 ?
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nop.i 0
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}
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{.mfb
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ld8 GR_COEFF_START= [ GR_TBL_START ] // Load pointer to coeff table
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nop.f 0
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(p12) br.cond.spnt SPECIAL_exp2 // Branch if nan, inf, zero
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}
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;;
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{.mlx
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setf.exp FR_OF_LIMIT= GR_OF_LIMIT // Set overflow limit
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movl GR_UF_LIMIT= 0xc4866000 // (-2^10-51) = -1075
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}
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;;
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{.mfi
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ldfpd FR_COEFF3, FR_COEFF4= [ GR_COEFF_START ], 16 // load C_3, C_4
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fma.s0 f8= f8, f1, f0 // normalize x
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nop.i 0
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}
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;;
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{.mmi
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setf.s FR_UF_LIMIT= GR_UF_LIMIT // Set underflow limit
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ldfe FR_COEFF1= [ GR_COEFF_START ], 16 // load C_1
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mov GR_EXP_CORR= 0xffff-126
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}
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;;
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{.mfi
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ldfe FR_COEFF2= [ GR_COEFF_START ], 16 // load C_2
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fma.s1 FR_KF0= f8, f1, FR_ROUNDVAL // y= x + 1.5*2^(63-10)
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nop.i 0
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}
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;;
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{.mfi
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mov GR_MASK= 1023
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fms.s1 FR_KF= FR_KF0, f1, FR_ROUNDVAL // (K+f)
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mov GR_MASK_low= 31
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}
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;;
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{.mfi
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getf.sig GR_KF0= FR_KF0 // (K+f)*2^10= round_to_int(y)
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fcmp.ge.s1 p12, p7= f8, FR_OF_LIMIT // x >= overflow threshold ?
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add GR_LOG_TBL= 256, GR_COEFF_START // Pointer to high T_table
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}
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;;
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{.mmi
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and GR_F_low= GR_KF0, GR_MASK_low // f_low
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and GR_F_high= GR_MASK, GR_KF0 // f_high*32
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shr GR_K= GR_KF0, 10 // K
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}
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;;
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{.mmi
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shladd GR_Flow_ADDR= GR_F_low, 3, GR_COEFF_START // address of 2^{f_low}
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add GR_BIAS= GR_K, GR_EXP_CORR // K= bias-2*63
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shr GR_Fh= GR_F_high, 5 // f_high
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}
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;;
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{.mfi
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setf.exp FR_2_TO_K= GR_BIAS // 2^{K-126}
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fnma.s1 FR_R= FR_KF, f1, f8 // r= x - (K+f)
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shladd GR_Fh_ADDR= GR_Fh, 3, GR_LOG_TBL // address of 2^{f_high}
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}
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{.mlx
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ldf8 FR_T_low= [ GR_Flow_ADDR ] // load T_low= 2^{f_low}
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movl GR_EMIN= 0xc47f8000 // EMIN= -1022
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}
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;;
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{.mfi
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ldf8 FR_T_high= [ GR_Fh_ADDR ] // load T_high= 2^{f_high}
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(p7) fcmp.lt.s1 p12, p7= f8, FR_UF_LIMIT // x<underflow threshold ?
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nop.i 0
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}
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;;
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{.mfi
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setf.s FR_EXPMIN= GR_EMIN // FR_EXPMIN= EMIN
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fma.s1 FR_P34= FR_COEFF4, FR_R, FR_COEFF3 // P34= C_3+C_4*r
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nop.i 0
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}
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{.mfb
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nop.m 0
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fma.s1 FR_R2= FR_R, FR_R, f0 // r*r
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(p12) br.cond.spnt OUT_RANGE_exp2
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}
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;;
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{.mfi
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nop.m 0
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fma.s1 FR_P12= FR_COEFF2, FR_R, FR_COEFF1 // P12= C_1+C_2*r
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nop.i 0
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}
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;;
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{.mfi
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nop.m 0
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fma.s1 FR_T_low_K= FR_T_low, FR_2_TO_K, f0 // T= 2^{K-126}*T_low
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nop.i 0
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}
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;;
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{.mfi
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nop.m 0
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fma.s1 FR_P14= FR_R2, FR_P34, FR_P12 // P14= P12+r2*P34
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nop.i 0
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}
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;;
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{.mfi
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nop.m 0
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fma.s1 FR_T= FR_T_low_K, FR_T_high, f0 // T= T*T_high
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nop.i 0
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}
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;;
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{.mfi
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nop.m 0
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fcmp.lt.s0 p6, p8= f8, FR_EXPMIN // underflow (x<EMIN) ?
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nop.i 0
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}
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;;
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{.mfi
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nop.m 0
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fma.s1 FR_P= FR_P14, FR_R, f0 // P= P14*r
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nop.i 0
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}
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;;
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{.mfb
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nop.m 0
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fma.d.s0 f8= FR_P, FR_T, FR_T // result= T+T*P
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(p8) br.ret.sptk b0 // return
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}
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;;
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{.mfb
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(p6) mov GR_Parameter_TAG= 162
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nop.f 0
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(p6) br.cond.sptk __libm_error_region
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}
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;;
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SPECIAL_exp2:
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{.mfi
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nop.m 0
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fclass.m p6, p0= f8, 0x22 // x= -Infinity ?
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nop.i 0
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}
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;;
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{.mfi
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nop.m 0
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fclass.m p7, p0= f8, 0x21 // x= +Infinity ?
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nop.i 0
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}
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;;
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{.mfi
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nop.m 0
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fclass.m p8, p0= f8, 0x7 // x= +/-Zero ?
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nop.i 0
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}
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{.mfb
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nop.m 0
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(p6) mov f8= f0 // exp2(-Infinity)= 0
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(p6) br.ret.spnt b0
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}
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;;
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{.mfb
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nop.m 0
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nop.f 0
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(p7) br.ret.spnt b0 // exp2(+Infinity)= +Infinity
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}
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;;
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{.mfb
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nop.m 0
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(p8) mov f8= f1 // exp2(+/-0)= 1
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(p8) br.ret.spnt b0
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}
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;;
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{.mfb
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nop.m 0
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fma.d.s0 f8= f8, f1, f0 // Remaining cases: NaNs
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br.ret.sptk b0
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}
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;;
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OUT_RANGE_exp2:
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// overflow: p8= 1
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{.mii
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(p8) mov GR_EXPMAX= 0x1fffe
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nop.i 0
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nop.i 0
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}
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;;
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{.mmb
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(p8) mov GR_Parameter_TAG= 161
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(p8) setf.exp FR_R= GR_EXPMAX
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nop.b 999
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}
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;;
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{.mfi
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nop.m 999
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(p8) fma.d.s0 f8= FR_R, FR_R, f0 // Create overflow
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nop.i 999
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}
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// underflow: p6= 1
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{.mii
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(p6) mov GR_Parameter_TAG= 162
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(p6) mov GR_EXPMAX= 1
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nop.i 0
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}
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;;
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{.mmb
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nop.m 0
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(p6) setf.exp FR_R= GR_EXPMAX
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nop.b 999
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}
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;;
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{.mfb
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nop.m 999
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(p6) fma.d.s0 f8= FR_R, FR_R, f0 // Create underflow
|
|
nop.b 0
|
|
}
|
|
;;
|
|
|
|
GLOBAL_LIBM_END(exp2)
|
|
|
|
|
|
LOCAL_LIBM_ENTRY(__libm_error_region)
|
|
|
|
.prologue
|
|
{.mfi
|
|
add GR_Parameter_Y= -32, sp // Parameter 2 value
|
|
nop.f 0
|
|
.save ar.pfs, GR_SAVE_PFS
|
|
mov GR_SAVE_PFS= ar.pfs // Save ar.pfs
|
|
}
|
|
|
|
{.mfi
|
|
.fframe 64
|
|
add sp= -64, sp // Create new stack
|
|
nop.f 0
|
|
mov GR_SAVE_GP= gp // Save gp
|
|
}
|
|
;;
|
|
|
|
{.mmi
|
|
stfd [ GR_Parameter_Y ]= FR_Y, 16 // STORE Parameter 2 on stack
|
|
add GR_Parameter_X= 16, sp // Parameter 1 address
|
|
.save b0, GR_SAVE_B0
|
|
mov GR_SAVE_B0= b0 // Save b0
|
|
}
|
|
;;
|
|
|
|
.body
|
|
{.mib
|
|
stfd [ GR_Parameter_X ]= FR_X // STORE Parameter 1 on stack
|
|
add GR_Parameter_RESULT= 0, GR_Parameter_Y // Parameter 3 address
|
|
nop.b 0
|
|
}
|
|
{.mib
|
|
stfd [ GR_Parameter_Y ]= FR_RESULT // STORE Parameter 3 on stack
|
|
add GR_Parameter_Y= -16, GR_Parameter_Y
|
|
br.call.sptk b0= __libm_error_support# // Call error handling function
|
|
}
|
|
;;
|
|
|
|
{.mmi
|
|
add GR_Parameter_RESULT= 48, sp
|
|
nop.m 0
|
|
nop.i 0
|
|
}
|
|
;;
|
|
|
|
{.mmi
|
|
ldfd f8= [ GR_Parameter_RESULT ] // Get return result off stack
|
|
.restore sp
|
|
add sp= 64, sp // Restore stack pointer
|
|
mov b0= GR_SAVE_B0 // Restore return address
|
|
}
|
|
;;
|
|
|
|
{.mib
|
|
mov gp= GR_SAVE_GP // Restore gp
|
|
mov ar.pfs= GR_SAVE_PFS // Restore ar.pfs
|
|
br.ret.sptk b0 // Return
|
|
}
|
|
;;
|
|
|
|
|
|
LOCAL_LIBM_END(__libm_error_region)
|
|
|
|
.type __libm_error_support#, @function
|
|
.global __libm_error_support#
|