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766 lines
29 KiB
ArmAsm
766 lines
29 KiB
ArmAsm
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.file "cbrtf.s"
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// Copyright (c) 2000 - 2003, 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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// 02/02/00 Initial version
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// 05/18/00 New version (modified algorithm)
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// 05/20/02 Cleaned up namespace and sf0 syntax
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// 01/28/03 Rescheduled some instructions for better performance
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// on Itanium 2, and reformatted
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//
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// API
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//==============================================================
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// float cbrtf(float)
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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 y= frcpa(a), where a is the argument
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//
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// cbrt(a)= cbrt(a*y)/cbrt(y) = cbrt(1 - (1-a*y)) * (1/cbrt(y))
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//
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// For all values of y, the 3 possible significands of 1/cbrt(y)
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// are stored in a table (T0) to 64 bits of accuracy. (There are
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// 3 possible significands because the exponent of y modulo 3
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// can be 0, 1, or 2.)
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//
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//
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// * cbrt(1 - (1-a*y)) is approximated by a degree-2 polynomial
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//
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// 1 - (1/3)*r - (1/9)*r^2
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//
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// in r = 1-a*y.
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//
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// The table values are stored for three exponent values and are
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// then multiplied by 2^(e/3) where e is the exponent of the input number.
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// This computation is carried out in parallel with the polynomial
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// evaluation:
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//
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// T= 2^(e/3) * T0
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//===============
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// input= x
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// C= frcpa(x)
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// r= 1 - C * x
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//
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// Special values
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//==============================================================
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// Registers used
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//==============================================================
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// p6, p7, p8, p12
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FR_R = f6
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FR_COEFF1 = f7
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FR_COEFF2 = f9
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FR_T0 = f10
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FR_T1 = f11
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FR_T2 = f12
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FR_2M63 = f13
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FR_ARG = f14
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FR_Y = f15
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GR_GP = r2
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GR_ADDR = r2
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GR_TMP5 = r3
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GR_CONST = r8
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GR_TMP63 = r8
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GR_SIGN = r9
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GR_CT2 = r10
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GR_CT3 = r11
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GR_TMP4 = r14
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GR_EBIAS3 = r15
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GR_REM = r16
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GR_SEXP = r17
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GR_2P63 = r18
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GR_SIGNIF = r19
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GR_I1 = r20
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GR_EBIAS = r21
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GR_EXP = r22
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GR_IT1 = r23
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GR_E5 = r24
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GR_IT1_3 = r25
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GR_TP1 = r26
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GR_TMP = r27
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GR_TMP2 = r28
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GR_TMP3 = r29
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GR_EXP3 = r30
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GR_ARGEXP = r31
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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 0xaaaab19b7e1f5ef9, 0x00003ffd // ~ 1/3
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data8 0xe38e5192a5a8e56c, 0x00003ffb // ~ 1/9
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LOCAL_OBJECT_END(poly_coeffs)
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// For every entry B in the frcpa table, this table contains
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// the significands of cbrt(1/B), cbrt(2/B), cbrt(4/B).
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// The index to this table is the same as the frcpa index.
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LOCAL_OBJECT_START(T_table)
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data8 0x80155c748c374836, 0xa160019ed37fb4ae
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data8 0xcb51ddcb9e93095e, 0x8040404b0879f7f9
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data8 0xa1960b5966da4608, 0xcb95f333968ad59b
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data8 0x806b5dce4b405c10, 0xa1cc5dbe6dc2aab4
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data8 0xcbda64292d3ffd97, 0x8096b586974669b1
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data8 0xa202f97995b69c0d, 0xcc1f3184af961596
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data8 0x80bcd273d952a028, 0xa232fe6eb0c0577d
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data8 0xcc5bb1ac954d33e2, 0x80e898c52813f2f3
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data8 0x8167c1dde03de7aa, 0xa30a5bd6e49e4ab8
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data8 0xa4115ce30548bc15, 0xceb666b2c347d1de
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data8 0x826097a62a8e5200, 0xa443df0e53df577a
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data8 0xcf763c47ee869f00, 0x82da06a527b18937
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data8 0xa4dcde37779adf4b, 0xcfb6cd3888d71785
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data8 0x8302e60b635ab394, 0xa5105d46152c938a
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data8 0xcff7aed4fbfbb447, 0x832bf8b2feec2f0e
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data8 0xa5441ce89825cb8d, 0xd038e1ce5167e3c6
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data8 0x83553f0ce00e276b, 0xa5781dad3e54d899
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data8 0xd07a66d7bfa0ebba, 0x837eb98b50f8322a
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data8 0xa5ac602406c4e68c, 0xd0bc3ea6b32d1b21
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data8 0x83a270f44c84f699, 0xa5d9601d95c2c0bc
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data8 0xd0f4f0e8f36c1bf8, 0x83cc4d7cfcfac5ca
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data8 0x83f65f78a8872b4c, 0xa6431f6e3fbd9658
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data8 0xd1f71682b2fa4575, 0x846fbd91b930bed2
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data8 0x88ca5eda67594784, 0xac5861d4aa441f0f
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|
|
data8 0xf935c88e0c7f419b, 0x9d225b983f6c1f96
|
||
|
|
data8 0xc5fa08f1ff20593c, 0xf96f5cd84fd86873
|
||
|
|
data8 0x9d3a9cca32261ed7, 0xc618980a79ce6862
|
||
|
|
data8 0xf995dd53ebdd9d6d, 0x9d52ecfccebe1768
|
||
|
|
data8 0xc6373a09e34b50fa, 0xf9bc75a034436a41
|
||
|
|
data8 0x9d77818d95b82f86, 0xc66550a6e0baaf35
|
||
|
|
data8 0xf9f686f26d5518de, 0x9d8ff7893fa4706c
|
||
|
|
data8 0xc6842241926342c9, 0xfa1d5b39b910a8c5
|
||
|
|
data8 0x9da87cbef36f2a5e, 0xc6a3070b7c93bb9e
|
||
|
|
data8 0xfa4447acc4ecbfd2, 0x9dcd6140b4a35aeb
|
||
|
|
data8 0xc6d18260bb84081b, 0xfa7ed7e51e6fdfb4
|
||
|
|
data8 0x9de60cd06dc6e2d4, 0xc6f0977c9416828b
|
||
|
|
data8 0xfaa601394d49a1a0, 0x9dfec7d4cc43b76f
|
||
|
|
data8 0xc70fc0117c641630, 0xfacd431644ce0e40
|
||
|
|
data8 0x9e17925ec9fccc4a, 0xc72efc34d7e615be
|
||
|
|
data8 0xfaf49d96f7a75909, 0x9e3cdf6db57dc075
|
||
|
|
data8 0xc75dfb441594141e, 0xfb2fd3c65e562fd5
|
||
|
|
data8 0x9e55d110b63637a8, 0xc77d68aa019bda4c
|
||
|
|
data8 0xfb576c5762024805, 0x9e6ed27594550d2e
|
||
|
|
data8 0xc79ce9ea478dbc4f, 0xfb7f1debc22c4040
|
||
|
|
data8 0x9e87e3adc385d393, 0xc7bc7f1ae453219d
|
||
|
|
data8 0xfba6e89f32d0190a, 0x9ead9b54b37a1055
|
||
|
|
data8 0xc7ec0476e15e141a, 0xfbe2c803a0894893
|
||
|
|
data8 0x9ec6d46a3d7de215, 0xc80bcbe16f1d540f
|
||
|
|
data8 0xfc0ad1ff0ed9ecf0, 0x9ee01d9108be3154
|
||
|
|
data8 0xc82ba78a5d349735, 0xfc32f57bdfbcbe7f
|
||
|
|
data8 0x9ef976db07288d04, 0xc84b978847a06b87
|
||
|
|
data8 0xfc5b32968f99b21c, 0x9f12e05a4759ec25
|
||
|
|
data8 0xc86b9bf1ee817bc6, 0xfc83896bc861ab08
|
||
|
|
data8 0x9f2c5a20f4da6668, 0xc88bb4de3667cdf4
|
||
|
|
data8 0xfcabfa1861ed4815, 0x9f52af78ed1733ca
|
||
|
|
data8 0xc8bc00e7fe9e23a3, 0xfce8d3cea7d3163e
|
||
|
|
data8 0x9f6c52426a39d003, 0xc8dc4d7ff2d25232
|
||
|
|
data8 0xfd118595143ee273, 0x9f860593d42fd7f3
|
||
|
|
data8 0xc8fcaeebcb40eb47, 0xfd3a519943d4865a
|
||
|
|
data8 0x9f9fc97fdb96bd51, 0xc91d25431426a663
|
||
|
|
data8 0xfd6337f8e1ae5a4b, 0x9fb99e194f4a7037
|
||
|
|
data8 0xc93db09d7fdb2949, 0xfd8c38d1c8e927eb
|
||
|
|
data8 0x9fd383731ca51db9, 0xc95e5112e721582a
|
||
|
|
data8 0xfdb5544205095a53, 0x9fed79a04fbf9423
|
||
|
|
data8 0xc97f06bb49787677, 0xfdde8a67d2613531
|
||
|
|
data8 0xa00780b413b24ee8, 0xc99fd1aecd6e1b06
|
||
|
|
data8 0xfe07db619e781611, 0xa02eab2c4474b0cd
|
||
|
|
data8 0xc9d12a3e27bb1625, 0xfe460768d80bf758
|
||
|
|
data8 0xa048dcd51ccfd142, 0xc9f22ad82ba3d5f0
|
||
|
|
data8 0xfe6f9bfb06cd32f6, 0xa0631fa894b11b8d
|
||
|
|
data8 0xca134113105e67b2, 0xfe994bcd3d14fcc2
|
||
|
|
data8 0xa07d73ba65e680af, 0xca346d07b045a876
|
||
|
|
data8 0xfec316fecaf3f2ab, 0xa097d91e6aaf71b0
|
||
|
|
data8 0xca55aecf0e94bb88, 0xfeecfdaf33fadb80
|
||
|
|
data8 0xa0b24fe89e02602f, 0xca77068257be9bab
|
||
|
|
data8 0xff16fffe2fa8fad6, 0xa0ccd82d1bd2f68b
|
||
|
|
data8 0xca98743ae1c693a8, 0xff411e0ba9db886d
|
||
|
|
data8 0xa0e77200215909e6, 0xcab9f8122c99a101
|
||
|
|
data8 0xff6b57f7c33e4e9a, 0xa1021d760d584855
|
||
|
|
data8 0xcadb9221e268c3b5, 0xff95ade2d1bd7358
|
||
|
|
data8 0xa11cdaa36068a57d, 0xcafd4283d8043dfd
|
||
|
|
data8 0xffc01fed60f86fb5, 0xa137a99cbd3f880b
|
||
|
|
data8 0xcb1f09520d37c6fb, 0xffeaae3832b63956
|
||
|
|
LOCAL_OBJECT_END(T_table)
|
||
|
|
|
||
|
|
|
||
|
|
|
||
|
|
|
||
|
|
|
||
|
|
|
||
|
|
|
||
|
|
.section .text
|
||
|
|
GLOBAL_LIBM_ENTRY(cbrtf)
|
||
|
|
|
||
|
|
|
||
|
|
{.mfi
|
||
|
|
getf.sig GR_SIGNIF = f8
|
||
|
|
// will continue only for normal/denormal numbers
|
||
|
|
fclass.nm.unc p12, p7 = f8, 0x1b
|
||
|
|
// GR_GP = pointer to C_1, C_2 followed by T_table
|
||
|
|
nop.i 0
|
||
|
|
}
|
||
|
|
{.mfi
|
||
|
|
addl GR_GP = @ltoff(poly_coeffs), gp
|
||
|
|
// normalize a
|
||
|
|
fma.s1 FR_ARG = f8, f1, f0
|
||
|
|
// GR_CT3 = bias-((2^8-1)/3) -63 = 0xffff-0x55-0x3f = 0xff6b
|
||
|
|
mov GR_CT3 = 0xff6b ;;
|
||
|
|
}
|
||
|
|
|
||
|
|
{.mmi
|
||
|
|
// get exponent
|
||
|
|
getf.exp GR_ARGEXP = f8
|
||
|
|
// load start address for C_1, C_2 followed by T_table
|
||
|
|
ld8 GR_ADDR = [ GR_GP ]
|
||
|
|
nop.i 0 ;;
|
||
|
|
}
|
||
|
|
|
||
|
|
{.mlx
|
||
|
|
// check if input significand is 0
|
||
|
|
(p7) cmp.eq p12, p7 = GR_SIGNIF, r0
|
||
|
|
// GR_2P63 = 2^63
|
||
|
|
movl GR_2P63 = 0x8000000000000000 ;;
|
||
|
|
}
|
||
|
|
|
||
|
|
{.mfi
|
||
|
|
nop.m 0
|
||
|
|
// y = frcpa(a)
|
||
|
|
// p7 = 1 for normal and denormal (but non-zero) arguments
|
||
|
|
(p7) frcpa.s0 FR_Y, p0 = f1, f8
|
||
|
|
// p9 = 1 if denormal input
|
||
|
|
cmp.gtu p9, p0 = GR_2P63, GR_SIGNIF
|
||
|
|
}
|
||
|
|
{.mfb
|
||
|
|
// load C_1
|
||
|
|
ldfe FR_COEFF1 = [ GR_ADDR ], 16
|
||
|
|
// if argument is 0, +/-Infinity, or NaN, return
|
||
|
|
(p12) fma.s.s0 f8 = f8, f1, f0
|
||
|
|
(p12) br.ret.spnt b0 ;;
|
||
|
|
}
|
||
|
|
|
||
|
|
{.mmi
|
||
|
|
// get normalized significand (for denormal inputs only)
|
||
|
|
(p9) getf.sig GR_SIGNIF = FR_ARG
|
||
|
|
// load C_2
|
||
|
|
ldfe FR_COEFF2 = [ GR_ADDR ], 16
|
||
|
|
// GR_CT2 = bias-(2^8-1)
|
||
|
|
mov GR_CT2 = 0xff00
|
||
|
|
}
|
||
|
|
|
||
|
|
{.mii
|
||
|
|
// get exponent (for denormal inputs only)
|
||
|
|
(p9) getf.exp GR_ARGEXP = FR_ARG
|
||
|
|
nop.i 0
|
||
|
|
mov GR_CONST = 0x20000 ;;
|
||
|
|
}
|
||
|
|
|
||
|
|
|
||
|
|
{.mii
|
||
|
|
// get GR_SIGN = sign
|
||
|
|
and GR_SIGN = GR_ARGEXP, GR_CONST
|
||
|
|
// eliminate leading 1 from GR_I1 = 1st table index
|
||
|
|
shl GR_I1 = GR_SIGNIF, 1
|
||
|
|
// eliminate sign from exponent
|
||
|
|
andcm GR_EBIAS = GR_ARGEXP, GR_CONST ;;
|
||
|
|
}
|
||
|
|
|
||
|
|
|
||
|
|
{.mfi
|
||
|
|
// subtract bias from GR_EXP = exponent
|
||
|
|
sub GR_EXP = GR_EBIAS, GR_CT2
|
||
|
|
// r = 1-a*y
|
||
|
|
fnma.s1 FR_R = FR_Y, FR_ARG, f1
|
||
|
|
// GR_IT1 = 1st table index (y_index8 bits)
|
||
|
|
shr.u GR_IT1 = GR_I1, 56 ;;
|
||
|
|
}
|
||
|
|
|
||
|
|
|
||
|
|
{.mii
|
||
|
|
// 1: exponent* = 5; // (2^{16}-1)/3 = 0x5555
|
||
|
|
shladd GR_E5 = GR_EXP, 2, GR_EXP
|
||
|
|
// GR_IT1_3 = 3*y_index
|
||
|
|
shladd GR_IT1_3 = GR_IT1, 1, GR_IT1
|
||
|
|
nop.i 0 ;;
|
||
|
|
}
|
||
|
|
|
||
|
|
|
||
|
|
{.mmi
|
||
|
|
// GR_TMP5 = (5*expon)*16+5*expon = (0x55)*expon
|
||
|
|
shladd GR_TMP5 = GR_E5, 4, GR_E5
|
||
|
|
// adjust T_table pointer by 1st index
|
||
|
|
shladd GR_TP1 = GR_IT1_3, 3, GR_ADDR
|
||
|
|
nop.i 0 ;;
|
||
|
|
}
|
||
|
|
|
||
|
|
|
||
|
|
{.mmi
|
||
|
|
// FR_T0 = T [ 0 ] [ y ]
|
||
|
|
ldf8 FR_T0 = [ GR_TP1 ], 8
|
||
|
|
// get 2^{-63}
|
||
|
|
mov GR_TMP63 = 0xffff + 63
|
||
|
|
// GR_TMP = (0x5500)*expon
|
||
|
|
shl GR_TMP = GR_TMP5, 8 ;;
|
||
|
|
}
|
||
|
|
|
||
|
|
|
||
|
|
{.mfi
|
||
|
|
// FR_T1 = T [ 1 ] [ y ]
|
||
|
|
ldf8 FR_T1 = [ GR_TP1 ], 8
|
||
|
|
// P_1 = C_1+C_2*r
|
||
|
|
fma.s1 FR_COEFF1 = FR_COEFF2, FR_R, FR_COEFF1
|
||
|
|
// GR_TMP2 = (0x5555)*expon
|
||
|
|
add GR_TMP2 = GR_TMP, GR_TMP5 ;;
|
||
|
|
}
|
||
|
|
|
||
|
|
|
||
|
|
{.mmi
|
||
|
|
// GR_TMP3 = (0x5556)*expon // 0x5556 = (2^{16}+2)/3
|
||
|
|
add GR_TMP3 = GR_TMP2, GR_EXP ;;
|
||
|
|
// FR_T2 = T [ 2 ] [ y ]
|
||
|
|
ldf8 FR_T2 = [ GR_TP1 ]
|
||
|
|
// GR_EXP3 = floor(expon/3)
|
||
|
|
shr GR_EXP3 = GR_TMP3, 16 ;;
|
||
|
|
}
|
||
|
|
|
||
|
|
|
||
|
|
{.mmi
|
||
|
|
setf.exp FR_2M63 = GR_TMP63
|
||
|
|
// GR_TMP4 = 3*exponent
|
||
|
|
shladd GR_TMP4 = GR_EXP3, 1, GR_EXP3
|
||
|
|
// bias exponent
|
||
|
|
add GR_EBIAS3 = GR_CT3, GR_EXP3 ;;
|
||
|
|
}
|
||
|
|
|
||
|
|
|
||
|
|
{.mmf
|
||
|
|
// get remainder of exponent/3
|
||
|
|
sub GR_REM = GR_EXP, GR_TMP4
|
||
|
|
// add sign to exponent
|
||
|
|
or GR_SEXP = GR_EBIAS3, GR_SIGN
|
||
|
|
// P_2 = -r*P_1
|
||
|
|
fnma.s1 FR_R = FR_COEFF1, FR_R, f0 ;;
|
||
|
|
}
|
||
|
|
|
||
|
|
|
||
|
|
|
||
|
|
{.mmi
|
||
|
|
// FR_ARG = sign*2^{exponent/3}
|
||
|
|
setf.exp FR_ARG = GR_SEXP
|
||
|
|
nop.m 0
|
||
|
|
// remainder = 0 ?
|
||
|
|
// p7=1 if input exponent is 3*j (remainder is 0)
|
||
|
|
cmp.eq.unc p7, p8 = r0, GR_REM ;;
|
||
|
|
}
|
||
|
|
|
||
|
|
|
||
|
|
{.mfi
|
||
|
|
// remainder = 1 ?
|
||
|
|
// p8=1 if input exponent is 3*j+1 (remainder is 1)
|
||
|
|
// p12=1 if input exponent is 3*j+2 (remainder is 2)
|
||
|
|
(p8) cmp.eq.unc p8, p12 = 1, GR_REM
|
||
|
|
// p7=1 -> remainder = 0 -> use T = FR_T0
|
||
|
|
(p7) fma.s1 f8 = FR_T0, FR_R, FR_T0
|
||
|
|
// argument is of the form 2^(3*k) ?
|
||
|
|
// ( GR_I1 holds significand bits, without the leading 1)
|
||
|
|
or GR_I1 = GR_I1, GR_REM ;;
|
||
|
|
}
|
||
|
|
|
||
|
|
|
||
|
|
.pred.rel "mutex", p12, p8
|
||
|
|
{.mfi
|
||
|
|
nop.m 0
|
||
|
|
// p8=1 -> remainder = 1 -> use FR_T1
|
||
|
|
(p8) fma.s1 f8 = FR_T1, FR_R, FR_T1
|
||
|
|
// argument is of the form 2^(3*k) ?
|
||
|
|
cmp.eq p14, p7 = GR_I1, r0
|
||
|
|
}
|
||
|
|
|
||
|
|
|
||
|
|
{.mfi
|
||
|
|
nop.m 0
|
||
|
|
// p12=1 -> remainder=2 -> result = T+T*P_2
|
||
|
|
(p12) fma.s1 f8 = FR_T2, FR_R, FR_T2
|
||
|
|
nop.i 0 ;;
|
||
|
|
}
|
||
|
|
|
||
|
|
|
||
|
|
.pred.rel "mutex", p14, p7
|
||
|
|
{.mfi
|
||
|
|
nop.m 0
|
||
|
|
// if argument is sgn*2^{3*(expon/3)}
|
||
|
|
(p14) fma.s.s0 f8 = FR_2M63, FR_ARG, f0
|
||
|
|
nop.i 0
|
||
|
|
}
|
||
|
|
{.mfb
|
||
|
|
nop.m 0
|
||
|
|
// T* = sgn*2^{expon/3}
|
||
|
|
(p7) fma.s.s0 f8 = f8, FR_ARG, f0
|
||
|
|
br.ret.sptk b0 ;;
|
||
|
|
}
|
||
|
|
|
||
|
|
|
||
|
|
GLOBAL_LIBM_END(cbrtf)
|
||
|
|
|
||
|
|
|
||
|
|
|
||
|
|
|