l_setox.s

VXWORKS源代码

/* l_setox.s - Motorola 68040 FP exponential routines (LIB) */

/* Copyright 1991-1993 Wind River Systems, Inc. */
	.data
	.globl	_copyright_wind_river
	.long	_copyright_wind_river

/*
modification history
--------------------
01f,12nov94,dvs  fixed clearcase conversion search/replace errors.
01e,21jul93,kdl  added .text (SPR #2372).
01d,23aug92,jcf  changed bxxx to jxx.
01c,26may92,rrr  the tree shuffle
01b,09jan92,kdl  added modification history; general cleanup.
01a,15aug91,kdl  original version, from Motorola FPSP v2.0.
*/

/*
DESCRIPTION


	setoxsa 3.1 12/10/90

	The entry point __l_setox computes the exponential of a value.
	__l_setoxd does the same except the input value is a denormalized
	number.	__l_setoxm1 computes exp(X)-1, and __l_setoxm1d computes
	exp(X)-1 for denormalized X.

	INPUT
	-----
	Double-extended value in memory location pointed to by address
	register a0.

	OUTPUT
	------
	exp(X) or exp(X)-1 returned in floating-point register fp0.

	ACCURACY and MONOTONICITY
	-------------------------
	The returned result is within 0.85 ulps in 64 significant bit, i.e.
	within 0.5001 ulp to 53 bits if the result is subsequently rounded
	to double precision. The result is provably monotonic in double
	precision.

	SPEED
	-----
	Two timings are measured, both in the copy-back mode. The
	first one is measured when the function is invoked the first time
	(so the instructions and data are not in cache), and the
	second one is measured when the function is reinvoked at the same
	input argument.

	The program __l_setox takes approximately 210/190 cycles for input
	argument X whose magnitude is less than 16380 log2, which
	is the usual situation.	For the less common arguments,
	depending on their values, the program may run faster or slower --
	but no worse than 10 slower even in the extreme cases.

	The program __l_setoxm1 takes approximately ???/??? cycles for input
	argument X, 0.25 <= |X| < 70log2. For |X| < 0.25, it takes
	approximately ???/??? cycles. For the less common arguments,
	depending on their values, the program may run faster or slower --
	but no worse than 10 slower even in the extreme cases.

	ALGORITHM and IMPLEMENTATION NOTES
	----------------------------------

	__l_setoxd
	------
	Step 1.	Set ans := 1.0

	Step 2.	Return	ans := ans + sign(X)*2^(-126). Exit.
	Notes:	This will always generate one exception -- inexact.


	__l_setox
	-----

	Step 1.	Filter out extreme cases of input argument.
		1.1	If |X| >= 2^(-65), go to Step 1.3.
		1.2	Go to Step 7.
		1.3	If |X| < 16380 log(2), go to Step 2.
		1.4	Go to Step 8.
	Notes:	The usual case should take the branches 1.1 -> 1.3 -> 2.
		 To avoid the use of floating-point comparisons, a
		 compact representation of |X| is used. This format is a
		 32-bit integer, the upper (more significant) 16 bits are
		 the sign and biased exponent field of |X||  the lower 16
		 bits are the 16 most significant fraction (including the
		 explicit bit) bits of |X|. Consequently, the comparisons
		 in Steps 1.1 and 1.3 can be performed by integer comparison.
		 Note also that the constant 16380 log(2) used in Step 1.3
		 is also in the compact form. Thus taking the branch
		 to Step 2 guarantees |X| < 16380 log(2). There is no harm
		 to have a small number of cases where |X| is less than,
		 but close to, 16380 log(2) and the branch to Step 9 is
		 taken.

	Step 2.	Calculate N = round-to-nearest-int( X * 64/log2 ).
		2.1	Set AdjFlag := 0 (indicates the branch 1.3 -> 2 was taken)
		2.2	N := round-to-nearest-integer( X * 64/log2 ).
		2.3	Calculate	J = N mod 64|  so J = 0,1,2,..., or 63.
		2.4	Calculate	M = (N - J)/64|  so N = 64M + J.
		2.5	Calculate the address of the stored value of 2^(J/64).
		2.6	Create the value Scale = 2^M.
	Notes:	The calculation in 2.2 is really performed by

			Z := X * constant
			N := round-to-nearest-integer(Z)

		 where

			constant := single-precision( 64/log 2 ).

		 Using a single-precision constant avoids memory access.
		 Another effect of using a single-precision "constant" is
		 that the calculated value Z is

			Z = X*(64/log2)*(1+eps), |eps| <= 2^(-24).

		 This error has to be considered later in Steps 3 and 4.

	Step 3.	Calculate X - N*log2/64.
		3.1	R := X + N*L1, where L1 := single-precision(-log2/64).
		3.2	R := R + N*L2, L2 := extended-precision(-log2/64 - L1).
	Notes:	a) The way L1 and L2 are chosen ensures L1+L2 approximate
		 the value	-log2/64	to 88 bits of accuracy.
		 b) N*L1 is exact because N is no longer than 22 bits and
		 L1 is no longer than 24 bits.
		 c) The calculation X+N*L1 is also exact due to cancellation.
		 Thus, R is practically X+N(L1+L2) to full 64 bits.
		 d) It is important to estimate how large can |R| be after
		 Step 3.2.

			N = rnd-to-int( X*64/log2 (1+eps) ), |eps|<=2^(-24)
			X*64/log2 (1+eps)	=	N + f,	|f| <= 0.5
			X*64/log2 - N	=	f - eps*X 64/log2
			X - N*log2/64	=	f*log2/64 - eps*X


		 Now |X| <= 16446 log2, thus

			|X - N*log2/64| <= (0.5 + 16446/2^(18))*log2/64
					<= 0.57 log2/64.
		 This bound will be used in Step 4.

	Step 4.	Approximate exp(R)-1 by a polynomial
			p = R + R*R*(A1 + R*(A2 + R*(A3 + R*(A4 + R*A5))))
	Notes:	a) In order to reduce memory access, the coefficients are
		 made as "short" as possible: A1 (which is 1/2), A4 and A5
		 are single precision|  A2 and A3 are double precision.
		 b) Even with the restrictions above,
			|p - (exp(R)-1)| < 2^(-68.8) for all |R| <= 0.0062.
		 Note that 0.0062 is slightly bigger than 0.57 log2/64.
		 c) To fully utilize the pipeline, p is separated into
		 two independent pieces of roughly equal complexities
			p = [ R + R*S*(A2 + S*A4) ]	+
				[ S*(A1 + S*(A3 + S*A5)) ]
		 where S = R*R.

	Step 5.	Compute 2^(J/64)*exp(R) = 2^(J/64)*(1+p) by
				ans := T + ( T*p + t)
		 where T and t are the stored values for 2^(J/64).
	Notes:	2^(J/64) is stored as T and t where T+t approximates
		 2^(J/64) to roughly 85 bits|  T is in extended precision
		 and t is in single precision. Note also that T is rounded
		 to 62 bits so that the last two bits of T are zero. The
		 reason for such a special form is that T-1, T-2, and T-8
		 will all be exact --- a property that will give much
		 more accurate computation of the function EXPM1.

	Step 6.	Reconstruction of exp(X)
			exp(X) = 2^M * 2^(J/64) * exp(R).
		6.1	If AdjFlag = 0, go to 6.3
		6.2	ans := ans * AdjScale
		6.3	Restore the user fpcr
		6.4	Return ans := ans * Scale. Exit.
	Notes:	If AdjFlag = 0, we have X = Mlog2 + Jlog2/64 + R,
		 |M| <= 16380, and Scale = 2^M. Moreover, exp(X) will
		 neither overflow nor underflow. If AdjFlag = 1, that
		 means that
			X = (M1+M)log2 + Jlog2/64 + R, |M1+M| >= 16380.
		 Hence, exp(X) may overflow or underflow or neither.
		 When that is the case, AdjScale = 2^(M1) where M1 is
		 approximately M. Thus 6.2 will never cause over/underflow.
		 Possible exception in 6.4 is overflow or underflow.
		 The inexact exception is not generated in 6.4. Although
		 one can argue that the inexact flag should always be
		 raised, to simulate that exception cost to much than the
		 flag is worth in practical uses.

	Step 7.	Return 1 + X.
		7.1	ans := X
		7.2	Restore user fpcr.
		7.3	Return ans := 1 + ans. Exit
	Notes:	For non-zero X, the inexact exception will always be
		 raised by 7.3. That is the only exception raised by 7.3.
		 Note also that we use the FMOVEM instruction to move X
		 in Step 7.1 to avoid unnecessary trapping. (Although
		 the FMOVEM may not seem relevant since X is normalized,
		 the precaution will be useful in the library version of
		 this code where the separate entry for denormalized inputs
		 will be done away with.)

	Step 8.	Handle exp(X) where |X| >= 16380log2.
		8.1	If |X| > 16480 log2, go to Step 9.
		(mimic 2.2 - 2.6)
		8.2	N := round-to-integer( X * 64/log2 )
		8.3	Calculate J = N mod 64, J = 0,1,...,63
		8.4	K := (N-J)/64, M1 := truncate(K/2), M = K-M1, AdjFlag := 1.
		8.5	Calculate the address of the stored value 2^(J/64).
		8.6	Create the values Scale = 2^M, AdjScale = 2^M1.
		8.7	Go to Step 3.
	Notes:	Refer to notes for 2.2 - 2.6.

	Step 9.	Handle exp(X), |X| > 16480 log2.
		9.1	If X < 0, go to 9.3
		9.2	ans := Huge, go to 9.4
		9.3	ans := Tiny.
		9.4	Restore user fpcr.
		9.5	Return ans := ans * ans. Exit.
	Notes:	Exp(X) will surely overflow or underflow, depending on
		 X's sign. "Huge" and "Tiny" are respectively large/tiny
		 extended-precision numbers whose square over/underflow
		 with an inexact result. Thus, 9.5 always raises the
		 inexact together with either overflow or underflow.


	__l_setoxm1d
	--------

	Step 1.	Set ans := 0

	Step 2.	Return	ans := X + ans. Exit.
	Notes:	This will return X with the appropriate rounding
		 precision prescribed by the user fpcr.

	__l_setoxm1
	-------

	Step 1.	Check |X|
		1.1	If |X| >= 1/4, go to Step 1.3.
		1.2	Go to Step 7.
		1.3	If |X| < 70 log(2), go to Step 2.
		1.4	Go to Step 10.
	Notes:	The usual case should take the branches 1.1 -> 1.3 -> 2.
		 However, it is conceivable |X| can be small very often
		 because EXPM1 is intended to evaluate exp(X)-1 accurately
		 when |X| is small. For further details on the comparisons,
		 see the notes on Step 1 of __l_setox.

	Step 2.	Calculate N = round-to-nearest-int( X * 64/log2 ).
		2.1	N := round-to-nearest-integer( X * 64/log2 ).
		2.2	Calculate	J = N mod 64|  so J = 0,1,2,..., or 63.
		2.3	Calculate	M = (N - J)/64|  so N = 64M + J.
		2.4	Calculate the address of the stored value of 2^(J/64).
		2.5	Create the values Sc = 2^M and OnebySc := -2^(-M).
	Notes:	See the notes on Step 2 of __l_setox.

	Step 3.	Calculate X - N*log2/64.
		3.1	R := X + N*L1, where L1 := single-precision(-log2/64).
		3.2	R := R + N*L2, L2 := extended-precision(-log2/64 - L1).
	Notes:	Applying the analysis of Step 3 of __l_setox in this case
		 shows that |R| <= 0.0055 (note that |X| <= 70 log2 in
		 this case).

	Step 4.	Approximate exp(R)-1 by a polynomial
			p = R+R*R*(A1+R*(A2+R*(A3+R*(A4+R*(A5+R*A6)))))
	Notes:	a) In order to reduce memory access, the coefficients are
		 made as "short" as possible: A1 (which is 1/2), A5 and A6
		 are single precision|  A2, A3 and A4 are double precision.
		 b) Even with the restriction above,
			|p - (exp(R)-1)| <	|R| * 2^(-72.7)
		 for all |R| <= 0.0055.
		 c) To fully utilize the pipeline, p is separated into
		 two independent pieces of roughly equal complexity
			p = [ R*S*(A2 + S*(A4 + S*A6)) ]	+
				[ R + S*(A1 + S*(A3 + S*A5)) ]
		 where S = R*R.

	Step 5.	Compute 2^(J/64)*p by
				p := T*p
		 where T and t are the stored values for 2^(J/64).
	Notes:	2^(J/64) is stored as T and t where T+t approximates
		 2^(J/64) to roughly 85 bits|  T is in extended precision
		 and t is in single precision. Note also that T is rounded
		 to 62 bits so that the last two bits of T are zero. The
		 reason for such a special form is that T-1, T-2, and T-8
		 will all be exact --- a property that will be exploited
		 in Step 6 below. The total relative error in p is no
		 bigger than 2^(-67.7) compared to the final result.

	Step 6.	Reconstruction of exp(X)-1
			exp(X)-1 = 2^M * ( 2^(J/64) + p - 2^(-M) ).
		6.1	If M <= 63, go to Step 6.3.
		6.2	ans := T + (p + (t + OnebySc)). Go to 6.6
		6.3	If M >= -3, go to 6.5.
		6.4	ans := (T + (p + t)) + OnebySc. Go to 6.6
		6.5	ans := (T + OnebySc) + (p + t).
		6.6	Restore user fpcr.
		6.7	Return ans := Sc * ans. Exit.
	Notes:	The various arrangements of the expressions give accurate
		 evaluations.

	Step 7.	exp(X)-1 for |X| < 1/4.
		7.1	If |X| >= 2^(-65), go to Step 9.
		7.2	Go to Step 8.

	Step 8.	Calculate exp(X)-1, |X| < 2^(-65).
		8.1	If |X| < 2^(-16312), goto 8.3
		8.2	Restore fpcr|  return ans := X - 2^(-16382). Exit.
		8.3	X := X * 2^(140).
		8.4	Restore fpcr|  ans := ans - 2^(-16382).
		 Return ans := ans*2^(140). Exit
	Notes:	The idea is to return "X - tiny" under the user
		 precision and rounding modes. To avoid unnecessary
		 inefficiency, we stay away from denormalized numbers the
		 best we can. For |X| >= 2^(-16312), the straightforward
		 8.2 generates the inexact exception as the case warrants.

	Step 9.	Calculate exp(X)-1, |X| < 1/4, by a polynomial
			p = X + X*X*(B1 + X*(B2 + |... + X*B12))
	Notes:	a) In order to reduce memory access, the coefficients are
		 made as "short" as possible: B1 (which is 1/2), B9 to B12
		 are single precision|  B3 to B8 are double precision|  and
		 B2 is double extended.
		 b) Even with the restriction above,
			|p - (exp(X)-1)| < |X| 2^(-70.6)
		 for all |X| <= 0.251.
		 Note that 0.251 is slightly bigger than 1/4.
		 c) To fully preserve accuracy, the polynomial is computed
		 as	X + ( S*B1 +	Q ) where S = X*X and
			Q	=	X*S*(B2 + X*(B3 + |... + X*B12))
		 d) To fully utilize the pipeline, Q is separated into
		 two independent pieces of roughly equal complexity
			Q = [ X*S*(B2 + S*(B4 + |... + S*B12)) ] +
				[ S*S*(B3 + S*(B5 + |... + S*B11)) ]

	Step 10.	Calculate exp(X)-1 for |X| >= 70 log 2.
		10.1 If X >= 70log2 , exp(X) - 1 = exp(X) for all practical
		 purposes. Therefore, go to Step 1 of __l_setox.
		10.2 If X <= -70log2, exp(X) - 1 = -1 for all practical purposes.
		 ans := -1
		 Restore user fpcr
		 Return ans := ans + 2^(-126). Exit.
	Notes:	10.2 will always create an inexact and return -1 + tiny
		 in the user rounding precision and mode.



		Copyright (C) Motorola, Inc. 1990
			All Rights Reserved

	THIS IS UNPUBLISHED PROPRIETARY SOURCE CODE OF MOTOROLA
	The copyright notice above does not evidence any
	actual or intended publication of such source code.

__l_setox	IDNT	2,1 Motorola 040 Floating Point Software Package

	section	8

NOMANUAL
*/

#include "fpsp040L.h"

L2:	.long	0x3FDC0000,0x82E30865,0x4361C4C6,0x00000000

EXPA3:	.long	0x3FA55555,0x55554431
EXPA2:	.long	0x3FC55555,0x55554018

HUGE:	.long	0x7FFE0000,0xFFFFFFFF,0xFFFFFFFF,0x00000000
TINY:	.long	0x00010000,0xFFFFFFFF,0xFFFFFFFF,0x00000000

EM1A4:	.long	0x3F811111,0x11174385
EM1A3:	.long	0x3FA55555,0x55554F5A

EM1A2:	.long	0x3FC55555,0x55555555,0x00000000,0x00000000

EM1B8:	.long	0x3EC71DE3,0xA5774682
EM1B7:	.long	0x3EFA01A0,0x19D7CB68

EM1B6:	.long	0x3F2A01A0,0x1A019DF3
EM1B5:	.long	0x3F56C16C,0x16C170E2

EM1B4:	.long	0x3F811111,0x11111111
EM1B3:	.long	0x3FA55555,0x55555555

EM1B2:	.long	0x3FFC0000,0xAAAAAAAA,0xAAAAAAAB
	.long	0x00000000

TWO140:	.long	0x48B00000,0x00000000
TWON140:	.long	0x37300000,0x00000000

EXPTBL:
	.long	0x3FFF0000,0x80000000,0x00000000,0x00000000
	.long	0x3FFF0000,0x8164D1F3,0xBC030774,0x9F841A9B
	.long	0x3FFF0000,0x82CD8698,0xAC2BA1D8,0x9FC1D5B9
	.long	0x3FFF0000,0x843A28C3,0xACDE4048,0xA0728369
	.long	0x3FFF0000,0x85AAC367,0xCC487B14,0x1FC5C95C
	.long	0x3FFF0000,0x871F6196,0x9E8D1010,0x1EE85C9F
	.long	0x3FFF0000,0x88980E80,0x92DA8528,0x9FA20729
	.long	0x3FFF0000,0x8A14D575,0x496EFD9C,0xA07BF9AF
	.long	0x3FFF0000,0x8B95C1E3,0xEA8BD6E8,0xA0020DCF
	.long	0x3FFF0000,0x8D1ADF5B,0x7E5BA9E4,0x205A63DA
	.long	0x3FFF0000,0x8EA4398B,0x45CD53C0,0x1EB70051
	.long	0x3FFF0000,0x9031DC43,0x1466B1DC,0x1F6EB029
	.long	0x3FFF0000,0x91C3D373,0xAB11C338,0xA0781494
	.long	0x3FFF0000,0x935A2B2F,0x13E6E92C,0x9EB319B0
	.long	0x3FFF0000,0x94F4EFA8,0xFEF70960,0x2017457D
	.long	0x3FFF0000,0x96942D37,0x20185A00,0x1F11D537
	.long	0x3FFF0000,0x9837F051,0x8DB8A970,0x9FB952DD
	.long	0x3FFF0000,0x99E04593,0x20B7FA64,0x1FE43087
	.long	0x3FFF0000,0x9B8D39B9,0xD54E5538,0x1FA2A818
	.long	0x3FFF0000,0x9D3ED9A7,0x2CFFB750,0x1FDE494D
	.long	0x3FFF0000,0x9EF53260,0x91A111AC,0x20504890
	.long	0x3FFF0000,0xA0B0510F,0xB9714FC4,0xA073691C
	.long	0x3FFF0000,0xA2704303,0x0C496818,0x1F9B7A05
	.long	0x3FFF0000,0xA43515AE,0x09E680A0,0xA0797126
	.long	0x3FFF0000,0xA5FED6A9,0xB15138EC,0xA071A140
	.long	0x3FFF0000,0xA7CD93B4,0xE9653568,0x204F62DA
	.long	0x3FFF0000,0xA9A15AB4,0xEA7C0EF8,0x1F283C4A
	.long	0x3FFF0000,0xAB7A39B5,0xA93ED338,0x9F9A7FDC
	.long	0x3FFF0000,0xAD583EEA,0x42A14AC8,0xA05B3FAC
	.long	0x3FFF0000,0xAF3B78AD,0x690A4374,0x1FDF2610
	.long	0x3FFF0000,0xB123F581,0xD2AC2590,0x9F705F90
	.long	0x3FFF0000,0xB311C412,0xA9112488,0x201F678A
	.long	0x3FFF0000,0xB504F333,0xF9DE6484,0x1F32FB13
	.long	0x3FFF0000,0xB6FD91E3,0x28D17790,0x20038B30
	.long	0x3FFF0000,0xB8FBAF47,0x62FB9EE8,0x200DC3CC
	.long	0x3FFF0000,0xBAFF5AB2,0x133E45FC,0x9F8B2AE6
	.long	0x3FFF0000,0xBD08A39F,0x580C36C0,0xA02BBF70
	.long	0x3FFF0000,0xBF1799B6,0x7A731084,0xA00BF518
	.long	0x3FFF0000,0xC12C4CCA,0x66709458,0xA041DD41
	.long	0x3FFF0000,0xC346CCDA,0x24976408,0x9FDF137B
	.long	0x3FFF0000,0xC5672A11,0x5506DADC,0x201F1568
	.long	0x3FFF0000,0xC78D74C8,0xABB9B15C,0x1FC13A2E
	.long	0x3FFF0000,0xC9B9BD86,0x6E2F27A4,0xA03F8F03
	.long	0x3FFF0000,0xCBEC14FE,0xF2727C5C,0x1FF4907D
	.long	0x3FFF0000,0xCE248C15,0x1F8480E4,0x9E6E53E4
	.long	0x3FFF0000,0xD06333DA,0xEF2B2594,0x1FD6D45C
	.long	0x3FFF0000,0xD2A81D91,0xF12AE45C,0xA076EDB9
	.long	0x3FFF0000,0xD4F35AAB,0xCFEDFA20,0x9FA6DE21
	.long	0x3FFF0000,0xD744FCCA,0xD69D6AF4,0x1EE69A2F
	.long	0x3FFF0000,0xD99D15C2,0x78AFD7B4,0x207F439F
	.long	0x3FFF0000,0xDBFBB797,0xDAF23754,0x201EC207