/* setox.s - Motorola 68040 FP exponential routines (EXC) *//* 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 shuffle01b,10jan92,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 __x_setox computes the exponential of a value.	__x_setoxd does the same except the input value is a denormalized	number.	__x_setoxm1 computes exp(X)-1, and __x_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 __x_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 __x_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	----------------------------------	__x_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.	__x_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.	__x_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.	__x_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 __x_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 __x_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 __x_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 __x_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.__x_setox	IDNT	2,1 Motorola 040 Floating Point Software Package	section	8NOMANUAL*/#include "fpsp040E.h"L2:	.long	0x3FDC0000,0x82E30865,0x4361C4C6,0x00000000EXPA3:	.long	0x3FA55555,0x55554431EXPA2:	.long	0x3FC55555,0x55554018HUGE:	.long	0x7FFE0000,0xFFFFFFFF,0xFFFFFFFF,0x00000000TINY:	.long	0x00010000,0xFFFFFFFF,0xFFFFFFFF,0x00000000EM1A4:	.long	0x3F811111,0x11174385EM1A3:	.long	0x3FA55555,0x55554F5AEM1A2:	.long	0x3FC55555,0x55555555,0x00000000,0x00000000EM1B8:	.long	0x3EC71DE3,0xA5774682EM1B7:	.long	0x3EFA01A0,0x19D7CB68EM1B6:	.long	0x3F2A01A0,0x1A019DF3EM1B5:	.long	0x3F56C16C,0x16C170E2EM1B4:	.long	0x3F811111,0x11111111EM1B3:	.long	0x3FA55555,0x55555555EM1B2:	.long	0x3FFC0000,0xAAAAAAAA,0xAAAAAAAB	.long	0x00000000TWO140:	.long	0x48B00000,0x00000000TWON140:	.long	0x37300000,0x00000000EXPTBL:	.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