path: root/arch/m68k/fpsp040/slogn.S

|
|	slogn.sa 3.1 12/10/90
|
|	slogn computes the natural logarithm of an
|	input value. slognd does the same except the input value is a
|	denormalized number. slognp1 computes log(1+X), and slognp1d
|	computes log(1+X) for denormalized X.
|
|	Input: Double-extended value in memory location pointed to by address
|		register a0.
|
|	Output:	log(X) or log(1+X) returned in floating-point register Fp0.
|
|	Accuracy and Monotonicity: The returned result is within 2 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: The program slogn takes approximately 190 cycles for input
|		argument X such that |X-1| >= 1/16, which is the usual
|		situation. For those arguments, slognp1 takes approximately
|		 210 cycles. For the less common arguments, the program will
|		 run no worse than 10% slower.
|
|	Algorithm:
|	LOGN:
|	Step 1. If |X-1| < 1/16, approximate log(X) by an odd polynomial in
|		u, where u = 2(X-1)/(X+1). Otherwise, move on to Step 2.
|
|	Step 2. X = 2**k * Y where 1 <= Y < 2. Define F to be the first seven
|		significant bits of Y plus 2**(-7), i.e. F = 1.xxxxxx1 in base
|		2 where the six "x" match those of Y. Note that |Y-F| <= 2**(-7).
|
|	Step 3. Define u = (Y-F)/F. Approximate log(1+u) by a polynomial in u,
|		log(1+u) = poly.
|
|	Step 4. Reconstruct log(X) = log( 2**k * Y ) = k*log(2) + log(F) + log(1+u)
|		by k*log(2) + (log(F) + poly). The values of log(F) are calculated
|		beforehand and stored in the program.
|
|	lognp1:
|	Step 1: If |X| < 1/16, approximate log(1+X) by an odd polynomial in
|		u where u = 2X/(2+X). Otherwise, move on to Step 2.
|
|	Step 2: Let 1+X = 2**k * Y, where 1 <= Y < 2. Define F as done in Step 2
|		of the algorithm for LOGN and compute log(1+X) as
|		k*log(2) + log(F) + poly where poly approximates log(1+u),
|		u = (Y-F)/F.
|
|	Implementation Notes:
|	Note 1. There are 64 different possible values for F, thus 64 log(F)'s
|		need to be tabulated. Moreover, the values of 1/F are also
|		tabulated so that the division in (Y-F)/F can be performed by a
|		multiplication.
|
|	Note 2. In Step 2 of lognp1, in order to preserved accuracy, the value
|		Y-F has to be calculated carefully when 1/2 <= X < 3/2.
|
|	Note 3. To fully exploit the pipeline, polynomials are usually separated
|		into two parts evaluated independently before being added up.
|

|		Copyright (C) Motorola, Inc. 1990
|			All Rights Reserved
|
|       For details on the license for this file, please see the
|       file, README, in this same directory.

|slogn	idnt	2,1 | Motorola 040 Floating Point Software Package

	|section	8

#include "fpsp.h"

BOUNDS1:  .long 0x3FFEF07D,0x3FFF8841
BOUNDS2:  .long 0x3FFE8000,0x3FFFC000

LOGOF2:	.long 0x3FFE0000,0xB17217F7,0xD1CF79AC,0x00000000

one:	.long 0x3F800000
zero:	.long 0x00000000
infty:	.long 0x7F800000
negone:	.long 0xBF800000

LOGA6:	.long 0x3FC2499A,0xB5E4040B
LOGA5:	.long 0xBFC555B5,0x848CB7DB

LOGA4:	.long 0x3FC99999,0x987D8730
LOGA3:	.long 0xBFCFFFFF,0xFF6F7E97

LOGA2:	.long 0x3FD55555,0x555555a4
LOGA1:	.long 0xBFE00000,0x00000008

LOGB5:	.long 0x3F175496,0xADD7DAD6
LOGB4:	.long 0x3F3C71C2,0xFE80C7E0

LOGB3:	.long 0x3F624924,0x928BCCFF
LOGB2:	.long 0x3F899999,0x999995EC

LOGB1:	.long 0x3FB55555,0x55555555
TWO:	.long 0x40000000,0x00000000

LTHOLD:	.long 0x3f990000,0x80000000,0x00000000,0x00000000

LOGTBL:
	.long  0x3FFE0000,0xFE03F80F,0xE03F80FE,0x00000000
	.long  0x3FF70000,0xFF015358,0x833C47E2,0x00000000
	.long  0x3FFE0000,0xFA232CF2,0x52138AC0,0x00000000
	.long  0x3FF90000,0xBDC8D83E,0xAD88D549,0x00000000
	.long  0x3FFE0000,0xF6603D98,0x0F6603DA,0x00000000
	.long  0x3FFA0000,0x9CF43DCF,0xF5EAFD48,0x00000000
	.long  0x3FFE0000,0xF2B9D648,0x0F2B9D65,0x00000000
	.long  0x3FFA0000,0xDA16EB88,0xCB8DF614,0x00000000
	.long  0x3FFE0000,0xEF2EB71F,0xC4345238,0x00000000
	.long  0x3FFB0000,0x8B29B775,0x1BD70743,0x00000000
	.long  0x3FFE0000,0xEBBDB2A5,0xC1619C8C,0x00000000
	.long  0x3FFB0000,0xA8D839F8,0x30C1FB49,0x00000000
	.long  0x3FFE0000,0xE865AC7B,0x7603A197,0x00000000
	.long  0x3FFB0000,0xC61A2EB1,0x8CD907AD,0x00000000
	.long  0x3FFE0000,0xE525982A,0xF70C880E,0x00000000
	.long  0x3FFB0000,0xE2F2A47A,0xDE3A18AF,0x00000000
	.long  0x3FFE0000,0xE1FC780E,0x1FC780E2,0x00000000
	.long  0x3FFB0000,0xFF64898E,0xDF55D551,0x00000000
	.long  0x3FFE0000,0xDEE95C4C,0xA037BA57,0x00000000
	.long  0x3FFC0000,0x8DB956A9,0x7B3D0148,0x00000000
	.long  0x3FFE0000,0xDBEB61EE,0xD19C5958,0x00000000
	.long  0x3FFC0000,0x9B8FE100,0xF47BA1DE,0x00000000
	.long  0x3FFE0000,0xD901B203,0x6406C80E,0x00000000
	.long  0x3FFC0000,0xA9372F1D,0x0DA1BD17,0x00000000
	.long  0x3FFE0000,0xD62B80D6,0x2B80D62C,0x00000000
	.long  0x3FFC0000,0xB6B07F38,0xCE90E46B,0x00000000
	.long  0x3FFE0000,0xD3680D36,0x80D3680D,0x00000000
	.long  0x3FFC0000,0xC3FD0329,0x06488481,0x00000000
	.long  0x3FFE0000,0xD0B69FCB,0xD2580D0B,0x00000000
	.long  0x3FFC0000,0xD11DE0FF,0x15AB18CA,0x00000000
	.long  0x3FFE0000,0xCE168A77,0x25080CE1,0x00000000
	.long  0x3FFC0000,0xDE1433A1,0x6C66B150,0x00000000
	.long  0x3FFE0000,0xCB8727C0,0x65C393E0,0x00000000
	.long  0x3FFC0000,0xEAE10B5A,0x7DDC8ADD,0x00000000
	.long  0x3FFE0000,0xC907DA4E,0x871146AD,0x00000000
	.long  0x3FFC0000,0xF7856E5E,0xE2C9B291,0x00000000
	.long  0x3FFE0000,0xC6980C69,0x80C6980C,0x00000000
	.long  0x3FFD0000,0x82012CA5,0xA68206D7,0x00000000
	.long  0x3FFE0000,0xC4372F85,0x5D824CA6,0x00000000
	.long  0x3FFD0000,0x882C5FCD,0x7256A8C5,0x00000000
	.long  0x3FFE0000,0xC1E4BBD5,0x95F6E947,0x00000000
	.long  0x3FFD0000,0x8E44C60B,0x4CCFD7DE,0x00000000
	.long  0x3FFE0000,0xBFA02FE8,0x0BFA02FF,0x00000000
	.long  0x3FFD0000,0x944AD09E,0xF4351AF6,0x00000000
	.long  0x3FFE0000,0xBD691047,0x07661AA3,0x00000000
	.long  0x3FFD0000,0x9A3EECD4,0xC3EAA6B2,0x00000000
	.long  0x3FFE0000,0xBB3EE721,0xA54D880C,0x00000000
	.long  0x3FFD0000,0xA0218434,0x353F1DE8,0x00000000
	.long  0x3FFE0000,0xB92143FA,0x36F5E02E,0x00000000
	.long  0x3FFD0000,0xA5F2FCAB,0xBBC506DA,0x00000000
	.long  0x3FFE0000,0xB70FBB5A,0x19BE3659,0x00000000
	.long  0x3FFD0000,0xABB3B8BA,0x2AD362A5,0x00000000
	.long  0x3FFE0000,0xB509E68A,0x9B94821F,0x00000000
	.long  0x3FFD0000,0xB1641795,0xCE3CA97B,0x00000000
	.long  0x3FFE0000,0xB30F6352,0x8917C80B,0x00000000
	.long  0x3FFD0000,0xB7047551,0x5D0F1C61,0x00000000
	.long  0x3FFE0000,0xB11FD3B8,0x0B11FD3C,0x00000000
	.long  0x3FFD0000,0xBC952AFE,0xEA3D13E1,0x00000000
	.long  0x3FFE0000,0xAF3ADDC6,0x80AF3ADE,0x00000000
	.long  0x3FFD0000,0xC2168ED0,0xF458BA4A,0x00000000
	.long  0x3FFE0000,0xAD602B58,0x0AD602B6,0x00000000
	.long  0x3FFD0000,0xC788F439,0xB3163BF1,0x00000000
	.long  0x3FFE0000,0xAB8F69E2,0x8359CD11,0x00000000
	.long  0x3FFD0000,0xCCECAC08,0xBF04565D,0x00000000
	.long  0x3FFE0000,0xA9C84A47,0xA07F5638,0x00000000
	.long  0x3FFD0000,0xD2420487,0x2DD85160,0x00000000
	.long  0x3FFE0000,0xA80A80A8,0x0A80A80B,0x00000000
	.long  0x3FFD0000,0xD7894992,0x3BC3588A,0x00000000
	.long  0x3FFE0000,0xA655C439,0x2D7B73A8,0x00000000
	.long  0x3FFD0000,0xDCC2C4B4,0x9887DACC,0x00000000
	.long  0x3FFE0000,0xA4A9CF1D,0x96833751,0x00000000
	.long  0x3FFD0000,0xE1EEBD3E,0x6D6A6B9E,0x00000000
	.long  0x3FFE0000,0xA3065E3F,0xAE7CD0E0,0x00000000
	.long  0x3FFD0000,0xE70D785C,0x2F9F5BDC,0x00000000
	.long  0x3FFE0000,0xA16B312E,0xA8FC377D,0x00000000
	.long  0x3FFD0000,0xEC1F392C,0x5179F283,0x00000000
	.long  0x3FFE0000,0x9FD809FD,0x809FD80A,0x00000000
	.long  0x3FFD0000,0xF12440D3,0xE36130E6,0x00000000
	.long  0x3FFE0000,0x9E4CAD23,0xDD5F3A20,0x00000000
	.long  0x3FFD0000,0xF61CCE92,0x346600BB,0x00000000
	.long  0x3FFE0000,0x9CC8E160,0xC3FB19B9,0x00000000
	.long  0x3FFD0000,0xFB091FD3,0x8145630A,0x00000000
	.long  0x3FFE0000,0x9B4C6F9E,0xF03A3CAA,0x00000000
	.long  0x3FFD0000,0xFFE97042,0xBFA4C2AD,0x00000000
	.long  0x3FFE0000,0x99D722DA,0xBDE58F06,0x00000000
	.long  0x3FFE0000,0x825EFCED,0x49369330,0x00000000
	.long  0x3FFE0000,0x9868C809,0x868C8098,0x00000000
	.long  0x3FFE0000,0x84C37A7A,0xB9A905C9,0x00000000
	.long  0x3FFE0000,0x97012E02,0x5C04B809,0x00000000
	.long  0x3FFE0000,0x87224C2E,0x8E645FB7,0x00000000
	.long  0x3FFE0000,0x95A02568,0x095A0257,0x00000000
	.long  0x3FFE0000,0x897B8CAC,0x9F7DE298,0x00000000
	.long  0x3FFE0000,0x94458094,0x45809446,0x00000000
	.long  0x3FFE0000,0x8BCF55DE,0xC4CD05FE,0x00000000
	.long  0x3FFE0000,0x92F11384,0x0497889C,0x00000000
	.long  0x3FFE0000,0x8E1DC0FB,0x89E125E5,0x00000000
	.long  0x3FFE0000,0x91A2B3C4,0xD5E6F809,0x00000000
	.long  0x3FFE0000,0x9066E68C,0x955B6C9B,0x00000000
	.long  0x3FFE0000,0x905A3863,0x3E06C43B,0x00000000
	.long  0x3FFE0000,0x92AADE74,0xC7BE59E0,0x00000000
	.long  0x3FFE0000,0x8F1779D9,0xFDC3A219,0x00000000
	.long  0x3FFE0000,0x94E9BFF6,0x15845643,0x00000000
	.long  0x3FFE0000,0x8DDA5202,0x37694809,0x00000000
	.long  0x3FFE0000,0x9723A1B7,0x20134203,0x00000000
	.long  0x3FFE0000,0x8CA29C04,0x6514E023,0x00000000
	.long  0x3FFE0000,0x995899C8,0x90EB8990,0x00000000
	.long  0x3FFE0000,0x8B70344A,0x139BC75A,0x00000000
	.long  0x3FFE0000,0x9B88BDAA,0x3A3DAE2F,0x00000000
	.long  0x3FFE0000,0x8A42F870,0x5669DB46,0x00000000
	.long  0x3FFE0000,0x9DB4224F,0xFFE1157C,0x00000000
	.long  0x3FFE0000,0x891AC73A,0xE9819B50,0x00000000
	.long  0x3FFE0000,0x9FDADC26,0x8B7A12DA,0x00000000
	.long  0x3FFE0000,0x87F78087,0xF78087F8,0x00000000
	.long  0x3FFE0000,0xA1FCFF17,0xCE733BD4,0x00000000
	.long  0x3FFE0000,0x86D90544,0x7A34ACC6,0x00000000
	.long  0x3FFE0000,0xA41A9E8F,0x5446FB9F,0x00000000
	.long  0x3FFE0000,0x85BF3761,0x2CEE3C9B,0x00000000
	.long  0x3FFE0000,0xA633CD7E,0x6771CD8B,0x00000000
	.long  0x3FFE0000,0x84A9F9C8,0x084A9F9D,0x00000000
	.long  0x3FFE0000,0xA8489E60,0x0B435A5E,0x00000000
	.long  0x3FFE0000,0x83993052,0x3FBE3368,0x00000000
	.long  0x3FFE0000,0xAA59233C,0xCCA4BD49,0x00000000
	.long  0x3FFE0000,0x828CBFBE,0xB9A020A3,0x00000000
	.long  0x3FFE0000,0xAC656DAE,0x6BCC4985,0x00000000
	.long  0x3FFE0000,0x81848DA8,0xFAF0D277,0x00000000
	.long  0x3FFE0000,0xAE6D8EE3,0x60BB2468,0x00000000
	.long  0x3FFE0000,0x80808080,0x80808081,0x00000000
	.long  0x3FFE0000,0xB07197A2,0x3C46C654,0x00000000

	.set	ADJK,L_SCR1

	.set	X,FP_SCR1
	.set	XDCARE,X+2
	.set	XFRAC,X+4

	.set	F,FP_SCR2
	.set	FFRAC,F+4

	.set	KLOG2,FP_SCR3

	.set	SAVEU,FP_SCR4

	| xref	t_frcinx
	|xref	t_extdnrm
	|xref	t_operr
	|xref	t_dz

	.global	slognd
slognd:
|--ENTRY POINT FOR LOG(X) FOR DENORMALIZED INPUT

	movel		#-100,ADJK(%a6)	| ...INPUT = 2^(ADJK) * FP0

|----normalize the input value by left shifting k bits (k to be determined
|----below), adjusting exponent and storing -k to  ADJK
|----the value TWOTO100 is no longer needed.
|----Note that this code assumes the denormalized input is NON-ZERO.

     moveml	%d2-%d7,-(%a7)		| ...save some registers
     movel	#0x00000000,%d3		| ...D3 is exponent of smallest norm. #
     movel	4(%a0),%d4
     movel	8(%a0),%d5		| ...(D4,D5) is (Hi_X,Lo_X)
     clrl	%d2			| ...D2 used for holding K

     tstl	%d4
     bnes	HiX_not0

HiX_0:
     movel	%d5,%d4
     clrl	%d5
     movel	#32,%d2
     clrl	%d6
     bfffo      %d4{#0:#32},%d6
     lsll      %d6,%d4
     addl	%d6,%d2			| ...(D3,D4,D5) is normalized

     movel	%d3,X(%a6)
     movel	%d4,XFRAC(%a6)
     movel	%d5,XFRAC+4(%a6)
     negl	%d2
     movel	%d2,ADJK(%a6)
     fmovex	X(%a6),%fp0
     moveml	(%a7)+,%d2-%d7		| ...restore registers
     lea	X(%a6),%a0
     bras	LOGBGN			| ...begin regular log(X)


HiX_not0:
     clrl	%d6
     bfffo	%d4{#0:#32},%d6		| ...find first 1
     movel	%d6,%d2			| ...get k
     lsll	%d6,%d4
     movel	%d5,%d7			| ...a copy of D5
     lsll	%d6,%d5
     negl	%d6
     addil	#32,%d6
     lsrl	%d6,%d7
     orl	%d7,%d4			| ...(D3,D4,D5) normalized

     movel	%d3,X(%a6)
     movel	%d4,XFRAC(%a6)
     movel	%d5,XFRAC+4(%a6)
     negl	%d2
     movel	%d2,ADJK(%a6)
     fmovex	X(%a6),%fp0
     moveml	(%a7)+,%d2-%d7		| ...restore registers
     lea	X(%a6),%a0
     bras	LOGBGN			| ...begin regular log(X)


	.global	slogn
slogn:
|--ENTRY POINT FOR LOG(X) FOR X FINITE, NON-ZERO, NOT NAN'S

	fmovex		(%a0),%fp0	| ...LOAD INPUT
	movel		#0x00000000,ADJK(%a6)

LOGBGN:
|--FPCR SAVED AND CLEARED, INPUT IS 2^(ADJK)*FP0, FP0 CONTAINS
|--A FINITE, NON-ZERO, NORMALIZED NUMBER.

	movel	(%a0),%d0
	movew	4(%a0),%d0

	movel	(%a0),X(%a6)
	movel	4(%a0),X+4(%a6)
	movel	8(%a0),X+8(%a6)

	cmpil	#0,%d0		| ...CHECK IF X IS NEGATIVE
	blt	LOGNEG		| ...LOG OF NEGATIVE ARGUMENT IS INVALID
	cmp2l	BOUNDS1,%d0	| ...X IS POSITIVE, CHECK IF X IS NEAR 1
	bcc	LOGNEAR1	| ...BOUNDS IS ROUGHLY [15/16, 17/16]

LOGMAIN:
|--THIS SHOULD BE THE USUAL CASE, X NOT VERY CLOSE TO 1

|--X = 2^(K) * Y, 1 <= Y < 2. THUS, Y = 1.XXXXXXXX....XX IN BINARY.
|--WE DEFINE F = 1.XXXXXX1, I.E. FIRST 7 BITS OF Y AND ATTACH A 1.
|--THE IDEA IS THAT LOG(X) = K*LOG2 + LOG(Y)
|--			 = K*LOG2 + LOG(F) + LOG(1 + (Y-F)/F).
|--NOTE THAT U = (Y-F)/F IS VERY SMALL AND THUS APPROXIMATING
|--LOG(1+U) CAN BE VERY EFFICIENT.
|--ALSO NOTE THAT THE VALUE 1/F IS STORED IN A TABLE SO THAT NO
|--DIVISION IS NEEDED TO CALCULATE (Y-F)/F.

|--GET K, Y, F, AND ADDRESS OF 1/F.
	asrl	#8,%d0
	asrl	#8,%d0		| ...SHIFTED 16 BITS, BIASED EXPO. OF X
	subil	#0x3FFF,%d0	| ...THIS IS K