.file "atan.s"
// Copyright (c) 2000 - 2003, Intel Corporation
// All rights reserved.
//
// Contributed 2000 by the Intel Numerics Group, Intel Corporation
//
// Redistribution and use in source and binary forms, with or without
// modification, are permitted provided that the following conditions are
// met:
//
// * Redistributions of source code must retain the above copyright
// notice, this list of conditions and the following disclaimer.
//
// * Redistributions in binary form must reproduce the above copyright
// notice, this list of conditions and the following disclaimer in the
// documentation and/or other materials provided with the distribution.
//
// * The name of Intel Corporation may not be used to endorse or promote
// products derived from this software without specific prior written
// permission.
// THIS SOFTWARE IS PROVIDED BY THE COPYRIGHT HOLDERS AND CONTRIBUTORS
// "AS IS" AND ANY EXPRESS OR IMPLIED WARRANTIES, INCLUDING, BUT NOT
// LIMITED TO, THE IMPLIED WARRANTIES OF MERCHANTABILITY AND FITNESS FOR
// A PARTICULAR PURPOSE ARE DISCLAIMED. IN NO EVENT SHALL INTEL OR ITS
// CONTRIBUTORS BE LIABLE FOR ANY DIRECT, INDIRECT, INCIDENTAL, SPECIAL,
// EXEMPLARY, OR CONSEQUENTIAL DAMAGES (INCLUDING, BUT NOT LIMITED TO,
// PROCUREMENT OF SUBSTITUTE GOODS OR SERVICES; LOSS OF USE, DATA, OR
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// OF LIABILITY, WHETHER IN CONTRACT, STRICT LIABILITY OR TORT (INCLUDING
// NEGLIGENCE OR OTHERWISE) ARISING IN ANY WAY OUT OF THE USE OF THIS
// SOFTWARE, EVEN IF ADVISED OF THE POSSIBILITY OF SUCH DAMAGE.
//
// Intel Corporation is the author of this code, and requests that all
// problem reports or change requests be submitted to it directly at
// http://www.intel.com/software/products/opensource/libraries/num.htm.
//
// History
//==============================================================
// 02/02/00 Initial version
// 04/13/00 Improved speed
// 04/19/00 Removed the qualifying predicate from the fmerge.s that
// takes the absolute value.
// 06/16/00 Reassigned FP registers to eliminate stalls on loads
// 08/30/00 Saved 5 cycles in main path by rearranging large argument logic
// and delaying use of result of fcmp in load by 1 group
// 05/20/02 Cleaned up namespace and sf0 syntax
// 08/20/02 Use atan2 algorithm with x=1 for better accuracy
// 02/06/03 Reordered header: .section, .global, .proc, .align
//
// API
//==============================================================
// double atan(double Y)
//
// Overview of operation
//==============================================================
//
// The atan function returns values in the interval [-pi/2,+pi/2].
//
// The algorithm used is the atan2(Y,X) algorithm where we fix X=1.0.
//
// There are two basic paths: swap true and swap false.
// atan2(Y,X) ==> atan2(V/U) where U >= V. If Y > X, we must swap.
//
// p6 swap True |Y| > |X|
// p7 swap False |Y| <= |X|
//
//
// Simple trigonometric identities show
// Region 1
// |Y|<=1.0, V=Y, U=1.0 atan2(Y,X) = sgnY * (0 + atan(V/U))
//
// Region 2
// |Y|>1.0, V=1.0, U=Y atan2(Y,X) = sgnY * (pi/2 - atan(V/U))
//
//
// We compute atan(V/U) from the identity
// atan(z) + atan([(V/U)-z] / [1+(V/U)z])
// where z is a limited precision approximation (16 bits) to V/U
//
// z is calculated with the assistance of the frcpa instruction.
//
// atan(z) is calculated by a polynomial z + z^3 * p(w), w=z^2
// where p(w) = P0+P1*w+...+P22*w^22
//
// Let d = [(V/U)-z] / [1+(V/U)z]) = (V-U*z)/(U+V*z)
//
// Approximate atan(d) by d + P0*d^3
// Let F = 1/(U+V*z) * (1-a), where |a|< 2^-8.8.
// Compute q(a) = 1 + a + ... + a^5.
// Then F*q(a) approximates the reciprocal to more than 50 bits.
// Special values
//==============================================================
// atan(QNAN) = QNAN
// atan(SNAN) = quieted SNAN
// atan(+-inf) = +- pi/2
// atan(+-0) = +-0
// Registers used
//==============================================================
// predicate registers used:
// p6 -> p15
// floating-point registers used:
// f8, input
// f32 -> f116
// general registers used
// r14 -> r16
// Assembly macros
//==============================================================
EXP_AD_P1 = r14
EXP_AD_P2 = r15
rsig_near_one = r16
atan2_Y = f8
atan2_X = f1
atan2_u1_X = f32
atan2_u1_Y = f33
atan2_z2_X = f34
atan2_two = f36
atan2_B1sq_Y = f37
atan2_z1_X = f38
atan2_B1X = f40
atan2_B1Y = f41
atan2_wp_X = f42
atan2_B1sq_X = f43
atan2_z = f44
atan2_w = f45
atan2_P0 = f46
atan2_P1 = f47
atan2_P2 = f48
atan2_P3 = f49
atan2_P4 = f50
atan2_P5 = f51
atan2_P6 = f52
atan2_P7 = f53
atan2_P8 = f54
atan2_P9 = f55
atan2_P10 = f56
atan2_P11 = f57
atan2_P12 = f58
atan2_P13 = f59
atan2_P14 = f60
atan2_P15 = f61
atan2_P16 = f62
atan2_P17 = f63
atan2_P18 = f64
atan2_P19 = f65
atan2_P20 = f66
atan2_P21 = f67
atan2_P22 = f68
atan2_pi_by_2 = f69
atan2_sgn_pi_by_2 = f69
atan2_V13 = f70
atan2_W11 = f71
atan2_E = f72
atan2_wp_Y = f73
atan2_V11 = f74
atan2_V12 = f75
atan2_V7 = f76
atan2_V8 = f77
atan2_W7 = f78
atan2_W8 = f79
atan2_W3 = f80
atan2_W4 = f81
atan2_V3 = f82
atan2_V4 = f83
atan2_F = f84
atan2_gV = f85
atan2_V10 = f86
atan2_zcub = f87
atan2_V6 = f88
atan2_V9 = f89
atan2_W10 = f90
atan2_W6 = f91
atan2_W2 = f92
atan2_V2 = f93
atan2_alpha = f94
atan2_alpha_1 = f95
atan2_gVF = f96
atan2_V5 = f97
atan2_W12 = f98
atan2_W5 = f99
atan2_alpha_sq = f100
atan2_Cp = f101
atan2_V1 = f102
atan2_ysq = f103
atan2_W1 = f104
atan2_alpha_cub = f105
atan2_C = f106
atan2_d = f108
atan2_A_hi = f109
atan2_dsq = f110
atan2_pd = f111
atan2_A_lo = f112
atan2_A = f113
atan2_Pp = f114
atan2_sgnY = f115
atan2_sig_near_one = f116
atan2_near_one = f116
/////////////////////////////////////////////////////////////
RODATA
.align 16
LOCAL_OBJECT_START(atan2_tb1)
data8 0xA21922DC45605EA1 , 0x00003FFA // P11
data8 0xB199DD6D2675C40F , 0x0000BFFA // P10
data8 0xC2F01E5DDD100DBE , 0x00003FFA // P9
data8 0xD78F28FC2A592781 , 0x0000BFFA // P8
data8 0xF0F03ADB3FC930D3 , 0x00003FFA // P7
data8 0x88887EBB209E3543 , 0x0000BFFB // P6
data8 0x9D89D7D55C3287A5 , 0x00003FFB // P5
data8 0xBA2E8B9793955C77 , 0x0000BFFB // P4
data8 0xE38E38E320A8A098 , 0x00003FFB // P3
data8 0x9249249247E37913 , 0x0000BFFC // P2
data8 0xCCCCCCCCCCC906CD , 0x00003FFC // P1
data8 0xAAAAAAAAAAAAA8A9 , 0x0000BFFD // P0
data8 0x0000000000000000 , 0x00000000 // pad to avoid bank conflict
LOCAL_OBJECT_END(atan2_tb1)
LOCAL_OBJECT_START(atan2_tb2)
data8 0xCE585A259BD8374C , 0x00003FF0 // P21
data8 0x9F90FB984D8E39D0 , 0x0000BFF3 // P20
data8 0x9D3436AABE218776 , 0x00003FF5 // P19
data8 0xDEC343E068A6D2A8 , 0x0000BFF6 // P18
data8 0xF396268151CFB11C , 0x00003FF7 // P17
data8 0xD818B4BB43D84BF2 , 0x0000BFF8 // P16
data8 0xA2270D30A90AA220 , 0x00003FF9 // P15
data8 0xD5F4F2182E7A8725 , 0x0000BFF9 // P14
data8 0x80D601879218B53A , 0x00003FFA // P13
data8 0x9297B23CCFFB291F , 0x0000BFFA // P12
data8 0xFE7E52D2A89995B3 , 0x0000BFEC // P22
data8 0xC90FDAA22168C235 , 0x00003FFF // pi/2
LOCAL_OBJECT_END(atan2_tb2)
.section .text
GLOBAL_LIBM_ENTRY(atan)
{ .mfi
nop.m 999
frcpa.s1 atan2_u1_Y,p7 = f1,atan2_Y
nop.i 999
}
{ .mfi
addl EXP_AD_P1 = @ltoff(atan2_tb1), gp
fma.s1 atan2_two = f1,f1,f1
nop.i 999
;;
}
{ .mfi
ld8 EXP_AD_P1 = [EXP_AD_P1]
frcpa.s1 atan2_u1_X,p6 = f1,atan2_X
nop.i 999
}
{ .mfi
nop.m 999
fma.s1 atan2_ysq = atan2_Y,atan2_Y,f0
nop.i 999
}
;;
{ .mfi
add EXP_AD_P2 = 0xd0,EXP_AD_P1
fmerge.s atan2_sgnY = atan2_Y,f1
nop.i 999
}
;;
{ .mfi
ldfe atan2_P11 = [EXP_AD_P1],16
fclass.m p10,p0 = atan2_Y, 0xc3 // Test for y=nan
nop.i 999
}
{ .mfi
ldfe atan2_P21 = [EXP_AD_P2],16
nop.f 999
nop.i 999
;;
}
{ .mfi
ldfe atan2_P10 = [EXP_AD_P1],16
fnma.s1 atan2_B1Y = atan2_u1_Y, atan2_Y, atan2_two
nop.i 999
}
{ .mfi
ldfe atan2_P20 = [EXP_AD_P2],16
fma.s1 atan2_wp_Y = atan2_u1_Y, atan2_u1_Y, f0
nop.i 999
;;
}
{ .mfi
ldfe atan2_P9 = [EXP_AD_P1],16
fma.s1 atan2_z1_X = atan2_u1_X, atan2_Y, f0
nop.i 999
}
{ .mfi
ldfe atan2_P19 = [EXP_AD_P2],16
fnma.s1 atan2_B1X = atan2_u1_X, atan2_X, atan2_two
nop.i 999
}
;;
{ .mfi
ldfe atan2_P8 = [EXP_AD_P1],16
fma.s1 atan2_z2_X = atan2_u1_X, atan2_ysq, f0
nop.i 999
}
{ .mfb
ldfe atan2_P18 = [EXP_AD_P2],16
(p10) fma.d.s0 f8 = atan2_Y,atan2_X,f0 // If y=nan, result quietized y
(p10) br.ret.spnt b0 // Exit if y=nan
}
;;
// p6 true if swap, means |y| > 1.0 or ysq > 1.0
// p7 true if no swap, means 1.0 >= |y| or 1.0 >= ysq
{ .mfi
ldfe atan2_P7 = [EXP_AD_P1],16
fcmp.ge.s1 p7,p6 = f1, atan2_ysq
nop.i 999
}
{ .mmf
ldfe atan2_P17 = [EXP_AD_P2],16
nop.m 999
nop.f 999
}
;;
{ .mfi
ldfe atan2_P6 = [EXP_AD_P1],16
fma.s1 atan2_E = atan2_u1_Y, atan2_B1Y, atan2_Y
nop.i 999
}
{ .mfi
ldfe atan2_P16 = [EXP_AD_P2],16
fma.s1 atan2_B1sq_Y = atan2_B1Y, atan2_B1Y, f0
nop.i 999
;;
}
{ .mfi
ldfe atan2_P5 = [EXP_AD_P1],16
(p7) fma.s1 atan2_wp_X = atan2_z1_X, atan2_z1_X, f0
nop.i 999
}
{ .mfi
ldfe atan2_P15 = [EXP_AD_P2],16
(p7) fma.s1 atan2_B1sq_X = atan2_B1X, atan2_B1X, f0
nop.i 999
;;
}
{ .mfi
ldfe atan2_P4 = [EXP_AD_P1],16
(p6) fma.s1 atan2_z = atan2_u1_Y, atan2_B1Y, f0
nop.i 999
}
{ .mfi
ldfe atan2_P14 = [EXP_AD_P2],16
(p7) fma.s1 atan2_E = atan2_z2_X, atan2_B1X, atan2_X
nop.i 999
;;
}
{ .mfi
ldfe atan2_P3 = [EXP_AD_P1],16
fcmp.eq.s0 p14,p15=atan2_X,atan2_Y // Dummy for denorm and invalid
nop.i 999
}
{ .mmf
ldfe atan2_P13 = [EXP_AD_P2],16
nop.m 999
(p7) fma.s1 atan2_z = atan2_z1_X, atan2_B1X, f0
;;
}
{ .mfi
ldfe atan2_P2 = [EXP_AD_P1],16
(p6) fma.s1 atan2_w = atan2_wp_Y, atan2_B1sq_Y,f0
nop.i 999
}
{ .mlx
ldfe atan2_P12 = [EXP_AD_P2],16
movl rsig_near_one = 0x8000000000000001 // signif near 1.0
;;
}
{ .mfi
ldfe atan2_P1 = [EXP_AD_P1],16
fclass.m p9,p0 = atan2_Y, 0x23 // test if y inf
nop.i 999
}
{ .mfi
ldfe atan2_P22 = [EXP_AD_P2],16
(p7) fma.s1 atan2_w = atan2_wp_X, atan2_B1sq_X,f0
nop.i 999
;;
}
{ .mfi
ldfe atan2_P0 = [EXP_AD_P1],16
frcpa.s1 atan2_F,p0 = f1, atan2_E
nop.i 999
}
{ .mfi
ldfe atan2_pi_by_2 = [EXP_AD_P2],16
(p6) fnma.s1 atan2_gV = atan2_Y, atan2_z, atan2_X
nop.i 999
;;
}
{ .mfi
setf.sig atan2_sig_near_one = rsig_near_one
(p7) fnma.s1 atan2_gV = atan2_X, atan2_z, atan2_Y
nop.i 999
}
{ .mfb
nop.m 999
(p9) fma.d.s0 f8 = atan2_sgnY, atan2_pi_by_2, f0 // +-pi/2 if y inf
(p9) br.ret.spnt b0 // exit if y inf, result is +-pi/2
;;
}
{ .mfi
nop.m 999
fma.s1 atan2_V13 = atan2_w, atan2_P11, atan2_P10
nop.i 999
}
{ .mfi
nop.m 999
fma.s1 atan2_W11 = atan2_w, atan2_P21, atan2_P20
nop.i 999
;;
}
{ .mfi
nop.m 999
fma.s1 atan2_V11 = atan2_w, atan2_P9, atan2_P8
nop.i 999
}
{ .mfi
nop.m 999
fma.s1 atan2_V12 = atan2_w, atan2_w, f0
nop.i 999
;;
}
{ .mfi
nop.m 999
fma.s1 atan2_V8 = atan2_w, atan2_P7 , atan2_P6
nop.i 999
}
{ .mfi
nop.m 999
fma.s1 atan2_W8 = atan2_w, atan2_P19, atan2_P18
nop.i 999
;;
}
{ .mfi
nop.m 999
fnma.s1 atan2_alpha = atan2_E, atan2_F, f1
nop.i 999
}
{ .mfi
nop.m 999
fnma.s1 atan2_alpha_1 = atan2_E, atan2_F, atan2_two
nop.i 999
;;
}
{ .mfi
nop.m 999
fma.s1 atan2_V7 = atan2_w, atan2_P5 , atan2_P4
nop.i 999
}
{ .mfi
nop.m 999
fma.s1 atan2_W7 = atan2_w, atan2_P17, atan2_P16
nop.i 999
;;
}
{ .mfi
nop.m 999
fma.s1 atan2_V4 = atan2_w, atan2_P3 , atan2_P2
nop.i 999
}
{ .mfi
nop.m 999
fma.s1 atan2_W4 = atan2_w, atan2_P15, atan2_P14
nop.i 999
;;
}
{ .mfi
nop.m 999
fma.s1 atan2_V3 = atan2_w, atan2_P1 , atan2_P0
nop.i 999
}
{ .mfi
nop.m 999
fma.s1 atan2_W3 = atan2_w, atan2_P13, atan2_P12
nop.i 999
;;
}
{ .mfi
nop.m 999
fma.s1 atan2_V10 = atan2_V12, atan2_V13, atan2_V11
nop.i 999
}
{ .mfi
nop.m 999
fma.s1 atan2_gVF = atan2_gV, atan2_F, f0
nop.i 999
;;
}
{ .mfi
nop.m 999
fma.s1 atan2_alpha_sq = atan2_alpha, atan2_alpha, f0
nop.i 999
}
{ .mfi
nop.m 999
fma.s1 atan2_Cp = atan2_alpha, atan2_alpha_1, f1
nop.i 999
;;
}
{ .mfi
nop.m 999
fma.s1 atan2_V9 = atan2_V12, atan2_V12, f0
nop.i 999
}
{ .mfi
nop.m 999
fma.s1 atan2_W10 = atan2_V12, atan2_P22 , atan2_W11
nop.i 999
;;
}
{ .mfi
nop.m 999
fma.s1 atan2_V6 = atan2_V12, atan2_V8 , atan2_V7
nop.i 999
}
{ .mfi
nop.m 999
fma.s1 atan2_W6 = atan2_V12, atan2_W8 , atan2_W7
nop.i 999
;;
}
{ .mfi
nop.m 999
fma.s1 atan2_V2 = atan2_V12, atan2_V4 , atan2_V3
nop.i 999
}
{ .mfi
nop.m 999
fma.s1 atan2_W2 = atan2_V12, atan2_W4 , atan2_W3
nop.i 999
;;
}
{ .mfi
nop.m 999
fma.s1 atan2_alpha_cub = atan2_alpha, atan2_alpha_sq, f0
nop.i 999
}
{ .mfi
nop.m 999
fma.s1 atan2_C = atan2_gVF, atan2_Cp, f0
nop.i 999
;;
}
{ .mfi
nop.m 999
fma.s1 atan2_W12 = atan2_V9, atan2_V9, f0
nop.i 999
;;
}
{ .mfi
nop.m 999
fma.s1 atan2_V5 = atan2_V9, atan2_V10, atan2_V6
nop.i 999
}
{ .mfi
nop.m 999
fma.s1 atan2_W5 = atan2_V9, atan2_W10, atan2_W6
nop.i 999
;;
}
{ .mfi
nop.m 999
fclass.m p8,p0 = atan2_Y, 0x07 // Test for y=0
nop.i 999
}
{ .mfi
nop.m 999
fma.s1 atan2_d = atan2_alpha_cub, atan2_C, atan2_C
nop.i 999
}
;;
{ .mfi
nop.m 999
fma.s1 atan2_W12 = atan2_V9, atan2_W12, f0
nop.i 999
}
;;
{ .mfi
nop.m 999
fma.s1 atan2_V1 = atan2_V9, atan2_V5, atan2_V2
nop.i 999
}
{ .mfi
nop.m 999
fma.s1 atan2_W1 = atan2_V9, atan2_W5, atan2_W2
nop.i 999
;;
}
{ .mfi
nop.m 999
(p8) fmerge.s f8 = atan2_sgnY, f0 // +-0 if y=0
nop.i 999
}
{ .mfb
nop.m 999
fma.s1 atan2_zcub = atan2_z, atan2_w, f0
(p8) br.ret.spnt b0 // Exit if y=0
;;
}
{ .mfi
nop.m 999
fma.s1 atan2_pd = atan2_P0, atan2_d, f0
nop.i 999
}
{ .mfi
nop.m 999
fma.s1 atan2_dsq = atan2_d, atan2_d, f0
nop.i 999
;;
}
{ .mfi
nop.m 999
fmerge.se atan2_near_one = f1, atan2_sig_near_one // Const ~1.0
nop.i 999
}
{ .mfi
nop.m 999
fma.s1 atan2_Pp = atan2_W12, atan2_W1, atan2_V1
nop.i 999
;;
}
{ .mfi
nop.m 999
fma.s1 atan2_sgn_pi_by_2 = atan2_pi_by_2, atan2_sgnY, f0
nop.i 999
}
{ .mfi
nop.m 999
fma.s1 atan2_A_lo = atan2_pd, atan2_dsq, atan2_d
nop.i 999
;;
}
{ .mfi
nop.m 999
fma.s1 atan2_A_hi = atan2_zcub, atan2_Pp, atan2_z
nop.i 999
;;
}
{ .mfi
nop.m 999
(p6) fma.s1 atan2_A = atan2_A_hi, f1, atan2_A_lo
nop.i 999
}
// For |Y| <= |X| and X > 0, result is A_hi + A_lo
{ .mfi
nop.m 999
(p7) fma.d.s0 f8 = atan2_A_hi, f1, atan2_A_lo
nop.i 999
;;
}
// For |Y| > |X|, result is +- pi/2 - (A_hi + A_lo)
// We perturb A by multiplying by 1.0+1ulp as we produce the result
// in order to get symmetrically rounded results in directed rounding modes.
// If we don't do this, there are a few cases where the trailing 11 bits of
// the significand of the result, before converting to double, are zero. These
// cases do not round symmetrically in round to +infinity or round to -infinity.
{ .mfb
nop.m 999
(p6) fnma.d.s0 f8 = atan2_A, atan2_near_one, atan2_sgn_pi_by_2
br.ret.sptk b0
;;
}
GLOBAL_LIBM_END(atan)
libm_alias_double_other (atan, atan)