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|
|       satan.sa 3.3 12/19/90
|
|       The entry point satan computes the arctangent of an
|       input value. satand does the same except the input value is a
|       denormalized number.
|
|       Input: Double-extended value in memory location pointed to by address
|               register a0.
|
|       Output: Arctan(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 satan takes approximately 160 cycles for input
|               argument X such that 1/16 < |X| < 16. For the other arguments,
|               the program will run no worse than 10% slower.
|
|       Algorithm:
|       Step 1. If |X| >= 16 or |X| < 1/16, go to Step 5.
|
|       Step 2. Let X = sgn * 2**k * 1.xxxxxxxx...x. Note that k = -4, -3,..., or 3.
|               Define F = sgn * 2**k * 1.xxxx1, i.e. the first 5 significant bits
|               of X with a bit-1 attached at the 6-th bit position. Define u
|               to be u = (X-F) / (1 + X*F).
|
|       Step 3. Approximate arctan(u) by a polynomial poly.
|
|       Step 4. Return arctan(F) + poly, arctan(F) is fetched from a table of values
|               calculated beforehand. Exit.
|
|       Step 5. If |X| >= 16, go to Step 7.
|
|       Step 6. Approximate arctan(X) by an odd polynomial in X. Exit.
|
|       Step 7. Define X' = -1/X. Approximate arctan(X') by an odd polynomial in X'.
|               Arctan(X) = sign(X)*Pi/2 + arctan(X'). Exit.
|
 
|               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|       THIS IS UNPUBLISHED PROPRIETARY SOURCE CODE OF MOTOROLA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.
 
|satan  idnt    2,1 | Motorola 040 Floating Point Software Package
 
        |section        8
 
        .include "fpsp.h"
 
BOUNDS1:        .long 0x3FFB8000,0x4002FFFF
 
ONE:    .long 0x3F800000
 
        .long 0x00000000
 
ATANA3: .long 0xBFF6687E,0x314987D8
ATANA2: .long 0x4002AC69,0x34A26DB3
 
ATANA1: .long 0xBFC2476F,0x4E1DA28E
ATANB6: .long 0x3FB34444,0x7F876989
 
ATANB5: .long 0xBFB744EE,0x7FAF45DB
ATANB4: .long 0x3FBC71C6,0x46940220
 
ATANB3: .long 0xBFC24924,0x921872F9
ATANB2: .long 0x3FC99999,0x99998FA9
 
ATANB1: .long 0xBFD55555,0x55555555
ATANC5: .long 0xBFB70BF3,0x98539E6A
 
ATANC4: .long 0x3FBC7187,0x962D1D7D
ATANC3: .long 0xBFC24924,0x827107B8
 
ATANC2: .long 0x3FC99999,0x9996263E
ATANC1: .long 0xBFD55555,0x55555536
 
PPIBY2: .long 0x3FFF0000,0xC90FDAA2,0x2168C235,0x00000000
NPIBY2: .long 0xBFFF0000,0xC90FDAA2,0x2168C235,0x00000000
PTINY:  .long 0x00010000,0x80000000,0x00000000,0x00000000
NTINY:  .long 0x80010000,0x80000000,0x00000000,0x00000000
 
ATANTBL:
        .long   0x3FFB0000,0x83D152C5,0x060B7A51,0x00000000
        .long   0x3FFB0000,0x8BC85445,0x65498B8B,0x00000000
        .long   0x3FFB0000,0x93BE4060,0x17626B0D,0x00000000
        .long   0x3FFB0000,0x9BB3078D,0x35AEC202,0x00000000
        .long   0x3FFB0000,0xA3A69A52,0x5DDCE7DE,0x00000000
        .long   0x3FFB0000,0xAB98E943,0x62765619,0x00000000
        .long   0x3FFB0000,0xB389E502,0xF9C59862,0x00000000
        .long   0x3FFB0000,0xBB797E43,0x6B09E6FB,0x00000000
        .long   0x3FFB0000,0xC367A5C7,0x39E5F446,0x00000000
        .long   0x3FFB0000,0xCB544C61,0xCFF7D5C6,0x00000000
        .long   0x3FFB0000,0xD33F62F8,0x2488533E,0x00000000
        .long   0x3FFB0000,0xDB28DA81,0x62404C77,0x00000000
        .long   0x3FFB0000,0xE310A407,0x8AD34F18,0x00000000
        .long   0x3FFB0000,0xEAF6B0A8,0x188EE1EB,0x00000000
        .long   0x3FFB0000,0xF2DAF194,0x9DBE79D5,0x00000000
        .long   0x3FFB0000,0xFABD5813,0x61D47E3E,0x00000000
        .long   0x3FFC0000,0x8346AC21,0x0959ECC4,0x00000000
        .long   0x3FFC0000,0x8B232A08,0x304282D8,0x00000000
        .long   0x3FFC0000,0x92FB70B8,0xD29AE2F9,0x00000000
        .long   0x3FFC0000,0x9ACF476F,0x5CCD1CB4,0x00000000
        .long   0x3FFC0000,0xA29E7630,0x4954F23F,0x00000000
        .long   0x3FFC0000,0xAA68C5D0,0x8AB85230,0x00000000
        .long   0x3FFC0000,0xB22DFFFD,0x9D539F83,0x00000000
        .long   0x3FFC0000,0xB9EDEF45,0x3E900EA5,0x00000000
        .long   0x3FFC0000,0xC1A85F1C,0xC75E3EA5,0x00000000
        .long   0x3FFC0000,0xC95D1BE8,0x28138DE6,0x00000000
        .long   0x3FFC0000,0xD10BF300,0x840D2DE4,0x00000000
        .long   0x3FFC0000,0xD8B4B2BA,0x6BC05E7A,0x00000000
        .long   0x3FFC0000,0xE0572A6B,0xB42335F6,0x00000000
        .long   0x3FFC0000,0xE7F32A70,0xEA9CAA8F,0x00000000
        .long   0x3FFC0000,0xEF888432,0x64ECEFAA,0x00000000
        .long   0x3FFC0000,0xF7170A28,0xECC06666,0x00000000
        .long   0x3FFD0000,0x812FD288,0x332DAD32,0x00000000
        .long   0x3FFD0000,0x88A8D1B1,0x218E4D64,0x00000000
        .long   0x3FFD0000,0x9012AB3F,0x23E4AEE8,0x00000000
        .long   0x3FFD0000,0x976CC3D4,0x11E7F1B9,0x00000000
        .long   0x3FFD0000,0x9EB68949,0x3889A227,0x00000000
        .long   0x3FFD0000,0xA5EF72C3,0x4487361B,0x00000000
        .long   0x3FFD0000,0xAD1700BA,0xF07A7227,0x00000000
        .long   0x3FFD0000,0xB42CBCFA,0xFD37EFB7,0x00000000
        .long   0x3FFD0000,0xBB303A94,0x0BA80F89,0x00000000
        .long   0x3FFD0000,0xC22115C6,0xFCAEBBAF,0x00000000
        .long   0x3FFD0000,0xC8FEF3E6,0x86331221,0x00000000
        .long   0x3FFD0000,0xCFC98330,0xB4000C70,0x00000000
        .long   0x3FFD0000,0xD6807AA1,0x102C5BF9,0x00000000
        .long   0x3FFD0000,0xDD2399BC,0x31252AA3,0x00000000
        .long   0x3FFD0000,0xE3B2A855,0x6B8FC517,0x00000000
        .long   0x3FFD0000,0xEA2D764F,0x64315989,0x00000000
        .long   0x3FFD0000,0xF3BF5BF8,0xBAD1A21D,0x00000000
        .long   0x3FFE0000,0x801CE39E,0x0D205C9A,0x00000000
        .long   0x3FFE0000,0x8630A2DA,0xDA1ED066,0x00000000
        .long   0x3FFE0000,0x8C1AD445,0xF3E09B8C,0x00000000
        .long   0x3FFE0000,0x91DB8F16,0x64F350E2,0x00000000
        .long   0x3FFE0000,0x97731420,0x365E538C,0x00000000
        .long   0x3FFE0000,0x9CE1C8E6,0xA0B8CDBA,0x00000000
        .long   0x3FFE0000,0xA22832DB,0xCADAAE09,0x00000000
        .long   0x3FFE0000,0xA746F2DD,0xB7602294,0x00000000
        .long   0x3FFE0000,0xAC3EC0FB,0x997DD6A2,0x00000000
        .long   0x3FFE0000,0xB110688A,0xEBDC6F6A,0x00000000
        .long   0x3FFE0000,0xB5BCC490,0x59ECC4B0,0x00000000
        .long   0x3FFE0000,0xBA44BC7D,0xD470782F,0x00000000
        .long   0x3FFE0000,0xBEA94144,0xFD049AAC,0x00000000
        .long   0x3FFE0000,0xC2EB4ABB,0x661628B6,0x00000000
        .long   0x3FFE0000,0xC70BD54C,0xE602EE14,0x00000000
        .long   0x3FFE0000,0xCD000549,0xADEC7159,0x00000000
        .long   0x3FFE0000,0xD48457D2,0xD8EA4EA3,0x00000000
        .long   0x3FFE0000,0xDB948DA7,0x12DECE3B,0x00000000
        .long   0x3FFE0000,0xE23855F9,0x69E8096A,0x00000000
        .long   0x3FFE0000,0xE8771129,0xC4353259,0x00000000
        .long   0x3FFE0000,0xEE57C16E,0x0D379C0D,0x00000000
        .long   0x3FFE0000,0xF3E10211,0xA87C3779,0x00000000
        .long   0x3FFE0000,0xF919039D,0x758B8D41,0x00000000
        .long   0x3FFE0000,0xFE058B8F,0x64935FB3,0x00000000
        .long   0x3FFF0000,0x8155FB49,0x7B685D04,0x00000000
        .long   0x3FFF0000,0x83889E35,0x49D108E1,0x00000000
        .long   0x3FFF0000,0x859CFA76,0x511D724B,0x00000000
        .long   0x3FFF0000,0x87952ECF,0xFF8131E7,0x00000000
        .long   0x3FFF0000,0x89732FD1,0x9557641B,0x00000000
        .long   0x3FFF0000,0x8B38CAD1,0x01932A35,0x00000000
        .long   0x3FFF0000,0x8CE7A8D8,0x301EE6B5,0x00000000
        .long   0x3FFF0000,0x8F46A39E,0x2EAE5281,0x00000000
        .long   0x3FFF0000,0x922DA7D7,0x91888487,0x00000000
        .long   0x3FFF0000,0x94D19FCB,0xDEDF5241,0x00000000
        .long   0x3FFF0000,0x973AB944,0x19D2A08B,0x00000000
        .long   0x3FFF0000,0x996FF00E,0x08E10B96,0x00000000
        .long   0x3FFF0000,0x9B773F95,0x12321DA7,0x00000000
        .long   0x3FFF0000,0x9D55CC32,0x0F935624,0x00000000
        .long   0x3FFF0000,0x9F100575,0x006CC571,0x00000000
        .long   0x3FFF0000,0xA0A9C290,0xD97CC06C,0x00000000
        .long   0x3FFF0000,0xA22659EB,0xEBC0630A,0x00000000
        .long   0x3FFF0000,0xA388B4AF,0xF6EF0EC9,0x00000000
        .long   0x3FFF0000,0xA4D35F10,0x61D292C4,0x00000000
        .long   0x3FFF0000,0xA60895DC,0xFBE3187E,0x00000000
        .long   0x3FFF0000,0xA72A51DC,0x7367BEAC,0x00000000
        .long   0x3FFF0000,0xA83A5153,0x0956168F,0x00000000
        .long   0x3FFF0000,0xA93A2007,0x7539546E,0x00000000
        .long   0x3FFF0000,0xAA9E7245,0x023B2605,0x00000000
        .long   0x3FFF0000,0xAC4C84BA,0x6FE4D58F,0x00000000
        .long   0x3FFF0000,0xADCE4A4A,0x606B9712,0x00000000
        .long   0x3FFF0000,0xAF2A2DCD,0x8D263C9C,0x00000000
        .long   0x3FFF0000,0xB0656F81,0xF22265C7,0x00000000
        .long   0x3FFF0000,0xB1846515,0x0F71496A,0x00000000
        .long   0x3FFF0000,0xB28AAA15,0x6F9ADA35,0x00000000
        .long   0x3FFF0000,0xB37B44FF,0x3766B895,0x00000000
        .long   0x3FFF0000,0xB458C3DC,0xE9630433,0x00000000
        .long   0x3FFF0000,0xB525529D,0x562246BD,0x00000000
        .long   0x3FFF0000,0xB5E2CCA9,0x5F9D88CC,0x00000000
        .long   0x3FFF0000,0xB692CADA,0x7ACA1ADA,0x00000000
        .long   0x3FFF0000,0xB736AEA7,0xA6925838,0x00000000
        .long   0x3FFF0000,0xB7CFAB28,0x7E9F7B36,0x00000000
        .long   0x3FFF0000,0xB85ECC66,0xCB219835,0x00000000
        .long   0x3FFF0000,0xB8E4FD5A,0x20A593DA,0x00000000
        .long   0x3FFF0000,0xB99F41F6,0x4AFF9BB5,0x00000000
        .long   0x3FFF0000,0xBA7F1E17,0x842BBE7B,0x00000000
        .long   0x3FFF0000,0xBB471285,0x7637E17D,0x00000000
        .long   0x3FFF0000,0xBBFABE8A,0x4788DF6F,0x00000000
        .long   0x3FFF0000,0xBC9D0FAD,0x2B689D79,0x00000000
        .long   0x3FFF0000,0xBD306A39,0x471ECD86,0x00000000
        .long   0x3FFF0000,0xBDB6C731,0x856AF18A,0x00000000
        .long   0x3FFF0000,0xBE31CAC5,0x02E80D70,0x00000000
        .long   0x3FFF0000,0xBEA2D55C,0xE33194E2,0x00000000
        .long   0x3FFF0000,0xBF0B10B7,0xC03128F0,0x00000000
        .long   0x3FFF0000,0xBF6B7A18,0xDACB778D,0x00000000
        .long   0x3FFF0000,0xBFC4EA46,0x63FA18F6,0x00000000
        .long   0x3FFF0000,0xC0181BDE,0x8B89A454,0x00000000
        .long   0x3FFF0000,0xC065B066,0xCFBF6439,0x00000000
        .long   0x3FFF0000,0xC0AE345F,0x56340AE6,0x00000000
        .long   0x3FFF0000,0xC0F22291,0x9CB9E6A7,0x00000000
 
        .set    X,FP_SCR1
        .set    XDCARE,X+2
        .set    XFRAC,X+4
        .set    XFRACLO,X+8
 
        .set    ATANF,FP_SCR2
        .set    ATANFHI,ATANF+4
        .set    ATANFLO,ATANF+8
 
 
        | xref  t_frcinx
        |xref   t_extdnrm
 
        .global satand
satand:
|--ENTRY POINT FOR ATAN(X) FOR DENORMALIZED ARGUMENT
 
        bra             t_extdnrm
 
        .global satan
satan:
|--ENTRY POINT FOR ATAN(X), HERE X IS FINITE, NON-ZERO, AND NOT NAN'S
 
        fmovex          (%a0),%fp0      | ...LOAD INPUT
 
        movel           (%a0),%d0
        movew           4(%a0),%d0
        fmovex          %fp0,X(%a6)
        andil           #0x7FFFFFFF,%d0
 
        cmpil           #0x3FFB8000,%d0         | ...|X| >= 1/16?
        bges            ATANOK1
        bra             ATANSM
 
ATANOK1:
        cmpil           #0x4002FFFF,%d0         | ...|X| < 16 ?
        bles            ATANMAIN
        bra             ATANBIG
 
 
|--THE MOST LIKELY CASE, |X| IN [1/16, 16). WE USE TABLE TECHNIQUE
|--THE IDEA IS ATAN(X) = ATAN(F) + ATAN( [X-F] / [1+XF] ).
|--SO IF F IS CHOSEN TO BE CLOSE TO X AND ATAN(F) IS STORED IN
|--A TABLE, ALL WE NEED IS TO APPROXIMATE ATAN(U) WHERE
|--U = (X-F)/(1+XF) IS SMALL (REMEMBER F IS CLOSE TO X). IT IS
|--TRUE THAT A DIVIDE IS NOW NEEDED, BUT THE APPROXIMATION FOR
|--ATAN(U) IS A VERY SHORT POLYNOMIAL AND THE INDEXING TO
|--FETCH F AND SAVING OF REGISTERS CAN BE ALL HIDED UNDER THE
|--DIVIDE. IN THE END THIS METHOD IS MUCH FASTER THAN A TRADITIONAL
|--ONE. NOTE ALSO THAT THE TRADITIONAL SCHEME THAT APPROXIMATE
|--ATAN(X) DIRECTLY WILL NEED TO USE A RATIONAL APPROXIMATION
|--(DIVISION NEEDED) ANYWAY BECAUSE A POLYNOMIAL APPROXIMATION
|--WILL INVOLVE A VERY LONG POLYNOMIAL.
 
|--NOW WE SEE X AS +-2^K * 1.BBBBBBB....B <- 1. + 63 BITS
|--WE CHOSE F TO BE +-2^K * 1.BBBB1
|--THAT IS IT MATCHES THE EXPONENT AND FIRST 5 BITS OF X, THE
|--SIXTH BITS IS SET TO BE 1. SINCE K = -4, -3, ..., 3, THERE
|--ARE ONLY 8 TIMES 16 = 2^7 = 128 |F|'S. SINCE ATAN(-|F|) IS
|-- -ATAN(|F|), WE NEED TO STORE ONLY ATAN(|F|).
 
ATANMAIN:
 
        movew           #0x0000,XDCARE(%a6)     | ...CLEAN UP X JUST IN CASE
        andil           #0xF8000000,XFRAC(%a6)  | ...FIRST 5 BITS
        oril            #0x04000000,XFRAC(%a6)  | ...SET 6-TH BIT TO 1
        movel           #0x00000000,XFRACLO(%a6)        | ...LOCATION OF X IS NOW F
 
        fmovex          %fp0,%fp1                       | ...FP1 IS X
        fmulx           X(%a6),%fp1             | ...FP1 IS X*F, NOTE THAT X*F > 0
        fsubx           X(%a6),%fp0             | ...FP0 IS X-F
        fadds           #0x3F800000,%fp1                | ...FP1 IS 1 + X*F
        fdivx           %fp1,%fp0                       | ...FP0 IS U = (X-F)/(1+X*F)
 
|--WHILE THE DIVISION IS TAKING ITS TIME, WE FETCH ATAN(|F|)
|--CREATE ATAN(F) AND STORE IT IN ATANF, AND
|--SAVE REGISTERS FP2.
 
        movel           %d2,-(%a7)      | ...SAVE d2 TEMPORARILY
        movel           %d0,%d2         | ...THE EXPO AND 16 BITS OF X
        andil           #0x00007800,%d0 | ...4 VARYING BITS OF F'S FRACTION
        andil           #0x7FFF0000,%d2 | ...EXPONENT OF F
        subil           #0x3FFB0000,%d2 | ...K+4
        asrl            #1,%d2
        addl            %d2,%d0         | ...THE 7 BITS IDENTIFYING F
        asrl            #7,%d0          | ...INDEX INTO TBL OF ATAN(|F|)
        lea             ATANTBL,%a1
        addal           %d0,%a1         | ...ADDRESS OF ATAN(|F|)
        movel           (%a1)+,ATANF(%a6)
        movel           (%a1)+,ATANFHI(%a6)
        movel           (%a1)+,ATANFLO(%a6)     | ...ATANF IS NOW ATAN(|F|)
        movel           X(%a6),%d0              | ...LOAD SIGN AND EXPO. AGAIN
        andil           #0x80000000,%d0 | ...SIGN(F)
        orl             %d0,ATANF(%a6)  | ...ATANF IS NOW SIGN(F)*ATAN(|F|)
        movel           (%a7)+,%d2      | ...RESTORE d2
 
|--THAT'S ALL I HAVE TO DO FOR NOW,
|--BUT ALAS, THE DIVIDE IS STILL CRANKING!
 
|--U IN FP0, WE ARE NOW READY TO COMPUTE ATAN(U) AS
|--U + A1*U*V*(A2 + V*(A3 + V)), V = U*U
|--THE POLYNOMIAL MAY LOOK STRANGE, BUT IS NEVERTHELESS CORRECT.
|--THE NATURAL FORM IS U + U*V*(A1 + V*(A2 + V*A3))
|--WHAT WE HAVE HERE IS MERELY  A1 = A3, A2 = A1/A3, A3 = A2/A3.
|--THE REASON FOR THIS REARRANGEMENT IS TO MAKE THE INDEPENDENT
|--PARTS A1*U*V AND (A2 + ... STUFF) MORE LOAD-BALANCED
 
 
        fmovex          %fp0,%fp1
        fmulx           %fp1,%fp1
        fmoved          ATANA3,%fp2
        faddx           %fp1,%fp2               | ...A3+V
        fmulx           %fp1,%fp2               | ...V*(A3+V)
        fmulx           %fp0,%fp1               | ...U*V
        faddd           ATANA2,%fp2     | ...A2+V*(A3+V)
        fmuld           ATANA1,%fp1     | ...A1*U*V
        fmulx           %fp2,%fp1               | ...A1*U*V*(A2+V*(A3+V))
 
        faddx           %fp1,%fp0               | ...ATAN(U), FP1 RELEASED
        fmovel          %d1,%FPCR               |restore users exceptions
        faddx           ATANF(%a6),%fp0 | ...ATAN(X)
        bra             t_frcinx
 
ATANBORS:
|--|X| IS IN d0 IN COMPACT FORM. FP1, d0 SAVED.
|--FP0 IS X AND |X| <= 1/16 OR |X| >= 16.
        cmpil           #0x3FFF8000,%d0
        bgt             ATANBIG | ...I.E. |X| >= 16
 
ATANSM:
|--|X| <= 1/16
|--IF |X| < 2^(-40), RETURN X AS ANSWER. OTHERWISE, APPROXIMATE
|--ATAN(X) BY X + X*Y*(B1+Y*(B2+Y*(B3+Y*(B4+Y*(B5+Y*B6)))))
|--WHICH IS X + X*Y*( [B1+Z*(B3+Z*B5)] + [Y*(B2+Z*(B4+Z*B6)] )
|--WHERE Y = X*X, AND Z = Y*Y.
 
        cmpil           #0x3FD78000,%d0
        blt             ATANTINY
|--COMPUTE POLYNOMIAL
        fmulx           %fp0,%fp0       | ...FP0 IS Y = X*X
 
 
        movew           #0x0000,XDCARE(%a6)
 
        fmovex          %fp0,%fp1
        fmulx           %fp1,%fp1               | ...FP1 IS Z = Y*Y
 
        fmoved          ATANB6,%fp2
        fmoved          ATANB5,%fp3
 
        fmulx           %fp1,%fp2               | ...Z*B6
        fmulx           %fp1,%fp3               | ...Z*B5
 
        faddd           ATANB4,%fp2     | ...B4+Z*B6
        faddd           ATANB3,%fp3     | ...B3+Z*B5
 
        fmulx           %fp1,%fp2               | ...Z*(B4+Z*B6)
        fmulx           %fp3,%fp1               | ...Z*(B3+Z*B5)
 
        faddd           ATANB2,%fp2     | ...B2+Z*(B4+Z*B6)
        faddd           ATANB1,%fp1     | ...B1+Z*(B3+Z*B5)
 
        fmulx           %fp0,%fp2               | ...Y*(B2+Z*(B4+Z*B6))
        fmulx           X(%a6),%fp0             | ...X*Y
 
        faddx           %fp2,%fp1               | ...[B1+Z*(B3+Z*B5)]+[Y*(B2+Z*(B4+Z*B6))]
 
 
        fmulx           %fp1,%fp0       | ...X*Y*([B1+Z*(B3+Z*B5)]+[Y*(B2+Z*(B4+Z*B6))])
 
        fmovel          %d1,%FPCR               |restore users exceptions
        faddx           X(%a6),%fp0
 
        bra             t_frcinx
 
ATANTINY:
|--|X| < 2^(-40), ATAN(X) = X
        movew           #0x0000,XDCARE(%a6)
 
        fmovel          %d1,%FPCR               |restore users exceptions
        fmovex          X(%a6),%fp0     |last inst - possible exception set
 
        bra             t_frcinx
 
ATANBIG:
|--IF |X| > 2^(100), RETURN     SIGN(X)*(PI/2 - TINY). OTHERWISE,
|--RETURN SIGN(X)*PI/2 + ATAN(-1/X).
        cmpil           #0x40638000,%d0
        bgt             ATANHUGE
 
|--APPROXIMATE ATAN(-1/X) BY
|--X'+X'*Y*(C1+Y*(C2+Y*(C3+Y*(C4+Y*C5)))), X' = -1/X, Y = X'*X'
|--THIS CAN BE RE-WRITTEN AS
|--X'+X'*Y*( [C1+Z*(C3+Z*C5)] + [Y*(C2+Z*C4)] ), Z = Y*Y.
 
        fmoves          #0xBF800000,%fp1        | ...LOAD -1
        fdivx           %fp0,%fp1               | ...FP1 IS -1/X
 
 
|--DIVIDE IS STILL CRANKING
 
        fmovex          %fp1,%fp0               | ...FP0 IS X'
        fmulx           %fp0,%fp0               | ...FP0 IS Y = X'*X'
        fmovex          %fp1,X(%a6)             | ...X IS REALLY X'
 
        fmovex          %fp0,%fp1
        fmulx           %fp1,%fp1               | ...FP1 IS Z = Y*Y
 
        fmoved          ATANC5,%fp3
        fmoved          ATANC4,%fp2
 
        fmulx           %fp1,%fp3               | ...Z*C5
        fmulx           %fp1,%fp2               | ...Z*B4
 
        faddd           ATANC3,%fp3     | ...C3+Z*C5
        faddd           ATANC2,%fp2     | ...C2+Z*C4
 
        fmulx           %fp3,%fp1               | ...Z*(C3+Z*C5), FP3 RELEASED
        fmulx           %fp0,%fp2               | ...Y*(C2+Z*C4)
 
        faddd           ATANC1,%fp1     | ...C1+Z*(C3+Z*C5)
        fmulx           X(%a6),%fp0             | ...X'*Y
 
        faddx           %fp2,%fp1               | ...[Y*(C2+Z*C4)]+[C1+Z*(C3+Z*C5)]
 
 
        fmulx           %fp1,%fp0               | ...X'*Y*([B1+Z*(B3+Z*B5)]
|                                       ...     +[Y*(B2+Z*(B4+Z*B6))])
        faddx           X(%a6),%fp0
 
        fmovel          %d1,%FPCR               |restore users exceptions
 
        btstb           #7,(%a0)
        beqs            pos_big
 
neg_big:
        faddx           NPIBY2,%fp0
        bra             t_frcinx
 
pos_big:
        faddx           PPIBY2,%fp0
        bra             t_frcinx
 
ATANHUGE:
|--RETURN SIGN(X)*(PIBY2 - TINY) = SIGN(X)*PIBY2 - SIGN(X)*TINY
        btstb           #7,(%a0)
        beqs            pos_huge
 
neg_huge:
        fmovex          NPIBY2,%fp0
        fmovel          %d1,%fpcr
        fsubx           NTINY,%fp0
        bra             t_frcinx
 
pos_huge:
        fmovex          PPIBY2,%fp0
        fmovel          %d1,%fpcr
        fsubx           PTINY,%fp0
        bra             t_frcinx
 
        |end

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[/] [or1k/] [trunk/] [rc203soc/] [sw/] [uClinux/] [arch/] [m68k/] [fpsp040/] [satan.S] - Blame information for rev 1765

Line No.	Rev	Author	Line
1	1623	jcastillo	`\|`
2			`\| satan.sa 3.3 12/19/90`
3			`\|`
4			`\| The entry point satan computes the arctangent of an`
5			`\| input value. satand does the same except the input value is a`
6			`\| denormalized number.`
7			`\|`
8			`\| Input: Double-extended value in memory location pointed to by address`
9			`\| register a0.`
10			`\|`
11			`\| Output: Arctan(X) returned in floating-point register Fp0.`
12			`\|`
13			`\| Accuracy and Monotonicity: The returned result is within 2 ulps in`
14			`\| 64 significant bit, i.e. within 0.5001 ulp to 53 bits if the`
15			`\| result is subsequently rounded to double precision. The`
16			`\| result is provably monotonic in double precision.`
17			`\|`
18			`\| Speed: The program satan takes approximately 160 cycles for input`
19			`\| argument X such that 1/16 < \|X\| < 16. For the other arguments,`
20			`\| the program will run no worse than 10% slower.`
21			`\|`
22			`\| Algorithm:`
23			`\| Step 1. If \|X\| >= 16 or \|X\| < 1/16, go to Step 5.`
24			`\|`
25			`\| Step 2. Let X = sgn * 2*k 1.xxxxxxxx...x. Note that k = -4, -3,..., or 3.`
26			`\| Define F = sgn * 2*k 1.xxxx1, i.e. the first 5 significant bits`
27			`\| of X with a bit-1 attached at the 6-th bit position. Define u`
28			`\| to be u = (X-F) / (1 + X*F).`
29			`\|`
30			`\| Step 3. Approximate arctan(u) by a polynomial poly.`
31			`\|`
32			`\| Step 4. Return arctan(F) + poly, arctan(F) is fetched from a table of values`
33			`\| calculated beforehand. Exit.`
34			`\|`
35			`\| Step 5. If \|X\| >= 16, go to Step 7.`
36			`\|`
37			`\| Step 6. Approximate arctan(X) by an odd polynomial in X. Exit.`
38			`\|`
39			`\| Step 7. Define X' = -1/X. Approximate arctan(X') by an odd polynomial in X'.`
40			`\| Arctan(X) = sign(X)*Pi/2 + arctan(X'). Exit.`
41			`\|`
42
43			`\| Copyright (C) Motorola, Inc. 1990`
44			`\| All Rights Reserved`
45			`\|`
46			`\| THIS IS UNPUBLISHED PROPRIETARY SOURCE CODE OF MOTOROLA`
47			`\| The copyright notice above does not evidence any`
48			`\| actual or intended publication of such source code.`
49
50			`\|satan idnt 2,1 \| Motorola 040 Floating Point Software Package`
51
52			`\|section 8`
53
54			`.include "fpsp.h"`
55
56			`BOUNDS1: .long 0x3FFB8000,0x4002FFFF`
57
58			`ONE: .long 0x3F800000`
59
60			`.long 0x00000000`
61
62			`ATANA3: .long 0xBFF6687E,0x314987D8`
63			`ATANA2: .long 0x4002AC69,0x34A26DB3`
64
65			`ATANA1: .long 0xBFC2476F,0x4E1DA28E`
66			`ATANB6: .long 0x3FB34444,0x7F876989`
67
68			`ATANB5: .long 0xBFB744EE,0x7FAF45DB`
69			`ATANB4: .long 0x3FBC71C6,0x46940220`
70
71			`ATANB3: .long 0xBFC24924,0x921872F9`
72			`ATANB2: .long 0x3FC99999,0x99998FA9`
73
74			`ATANB1: .long 0xBFD55555,0x55555555`
75			`ATANC5: .long 0xBFB70BF3,0x98539E6A`
76
77			`ATANC4: .long 0x3FBC7187,0x962D1D7D`
78			`ATANC3: .long 0xBFC24924,0x827107B8`
79
80			`ATANC2: .long 0x3FC99999,0x9996263E`
81			`ATANC1: .long 0xBFD55555,0x55555536`
82
83			`PPIBY2: .long 0x3FFF0000,0xC90FDAA2,0x2168C235,0x00000000`
84			`NPIBY2: .long 0xBFFF0000,0xC90FDAA2,0x2168C235,0x00000000`
85			`PTINY: .long 0x00010000,0x80000000,0x00000000,0x00000000`
86			`NTINY: .long 0x80010000,0x80000000,0x00000000,0x00000000`
87
88			`ATANTBL:`
89			`.long 0x3FFB0000,0x83D152C5,0x060B7A51,0x00000000`
90			`.long 0x3FFB0000,0x8BC85445,0x65498B8B,0x00000000`
91			`.long 0x3FFB0000,0x93BE4060,0x17626B0D,0x00000000`
92			`.long 0x3FFB0000,0x9BB3078D,0x35AEC202,0x00000000`
93			`.long 0x3FFB0000,0xA3A69A52,0x5DDCE7DE,0x00000000`
94			`.long 0x3FFB0000,0xAB98E943,0x62765619,0x00000000`
95			`.long 0x3FFB0000,0xB389E502,0xF9C59862,0x00000000`
96			`.long 0x3FFB0000,0xBB797E43,0x6B09E6FB,0x00000000`
97			`.long 0x3FFB0000,0xC367A5C7,0x39E5F446,0x00000000`
98			`.long 0x3FFB0000,0xCB544C61,0xCFF7D5C6,0x00000000`
99			`.long 0x3FFB0000,0xD33F62F8,0x2488533E,0x00000000`
100			`.long 0x3FFB0000,0xDB28DA81,0x62404C77,0x00000000`
101			`.long 0x3FFB0000,0xE310A407,0x8AD34F18,0x00000000`
102			`.long 0x3FFB0000,0xEAF6B0A8,0x188EE1EB,0x00000000`
103			`.long 0x3FFB0000,0xF2DAF194,0x9DBE79D5,0x00000000`
104			`.long 0x3FFB0000,0xFABD5813,0x61D47E3E,0x00000000`
105			`.long 0x3FFC0000,0x8346AC21,0x0959ECC4,0x00000000`
106			`.long 0x3FFC0000,0x8B232A08,0x304282D8,0x00000000`
107			`.long 0x3FFC0000,0x92FB70B8,0xD29AE2F9,0x00000000`
108			`.long 0x3FFC0000,0x9ACF476F,0x5CCD1CB4,0x00000000`
109			`.long 0x3FFC0000,0xA29E7630,0x4954F23F,0x00000000`
110			`.long 0x3FFC0000,0xAA68C5D0,0x8AB85230,0x00000000`
111			`.long 0x3FFC0000,0xB22DFFFD,0x9D539F83,0x00000000`
112			`.long 0x3FFC0000,0xB9EDEF45,0x3E900EA5,0x00000000`
113			`.long 0x3FFC0000,0xC1A85F1C,0xC75E3EA5,0x00000000`
114			`.long 0x3FFC0000,0xC95D1BE8,0x28138DE6,0x00000000`
115			`.long 0x3FFC0000,0xD10BF300,0x840D2DE4,0x00000000`
116			`.long 0x3FFC0000,0xD8B4B2BA,0x6BC05E7A,0x00000000`
117			`.long 0x3FFC0000,0xE0572A6B,0xB42335F6,0x00000000`
118			`.long 0x3FFC0000,0xE7F32A70,0xEA9CAA8F,0x00000000`
119			`.long 0x3FFC0000,0xEF888432,0x64ECEFAA,0x00000000`
120			`.long 0x3FFC0000,0xF7170A28,0xECC06666,0x00000000`
121			`.long 0x3FFD0000,0x812FD288,0x332DAD32,0x00000000`
122			`.long 0x3FFD0000,0x88A8D1B1,0x218E4D64,0x00000000`
123			`.long 0x3FFD0000,0x9012AB3F,0x23E4AEE8,0x00000000`
124			`.long 0x3FFD0000,0x976CC3D4,0x11E7F1B9,0x00000000`
125			`.long 0x3FFD0000,0x9EB68949,0x3889A227,0x00000000`
126			`.long 0x3FFD0000,0xA5EF72C3,0x4487361B,0x00000000`
127			`.long 0x3FFD0000,0xAD1700BA,0xF07A7227,0x00000000`
128			`.long 0x3FFD0000,0xB42CBCFA,0xFD37EFB7,0x00000000`
129			`.long 0x3FFD0000,0xBB303A94,0x0BA80F89,0x00000000`
130			`.long 0x3FFD0000,0xC22115C6,0xFCAEBBAF,0x00000000`
131			`.long 0x3FFD0000,0xC8FEF3E6,0x86331221,0x00000000`
132			`.long 0x3FFD0000,0xCFC98330,0xB4000C70,0x00000000`
133			`.long 0x3FFD0000,0xD6807AA1,0x102C5BF9,0x00000000`
134			`.long 0x3FFD0000,0xDD2399BC,0x31252AA3,0x00000000`
135			`.long 0x3FFD0000,0xE3B2A855,0x6B8FC517,0x00000000`
136			`.long 0x3FFD0000,0xEA2D764F,0x64315989,0x00000000`
137			`.long 0x3FFD0000,0xF3BF5BF8,0xBAD1A21D,0x00000000`
138			`.long 0x3FFE0000,0x801CE39E,0x0D205C9A,0x00000000`
139			`.long 0x3FFE0000,0x8630A2DA,0xDA1ED066,0x00000000`
140			`.long 0x3FFE0000,0x8C1AD445,0xF3E09B8C,0x00000000`
141			`.long 0x3FFE0000,0x91DB8F16,0x64F350E2,0x00000000`
142			`.long 0x3FFE0000,0x97731420,0x365E538C,0x00000000`
143			`.long 0x3FFE0000,0x9CE1C8E6,0xA0B8CDBA,0x00000000`
144			`.long 0x3FFE0000,0xA22832DB,0xCADAAE09,0x00000000`
145			`.long 0x3FFE0000,0xA746F2DD,0xB7602294,0x00000000`
146			`.long 0x3FFE0000,0xAC3EC0FB,0x997DD6A2,0x00000000`
147			`.long 0x3FFE0000,0xB110688A,0xEBDC6F6A,0x00000000`
148			`.long 0x3FFE0000,0xB5BCC490,0x59ECC4B0,0x00000000`
149			`.long 0x3FFE0000,0xBA44BC7D,0xD470782F,0x00000000`
150			`.long 0x3FFE0000,0xBEA94144,0xFD049AAC,0x00000000`
151			`.long 0x3FFE0000,0xC2EB4ABB,0x661628B6,0x00000000`
152			`.long 0x3FFE0000,0xC70BD54C,0xE602EE14,0x00000000`
153			`.long 0x3FFE0000,0xCD000549,0xADEC7159,0x00000000`
154			`.long 0x3FFE0000,0xD48457D2,0xD8EA4EA3,0x00000000`
155			`.long 0x3FFE0000,0xDB948DA7,0x12DECE3B,0x00000000`
156			`.long 0x3FFE0000,0xE23855F9,0x69E8096A,0x00000000`
157			`.long 0x3FFE0000,0xE8771129,0xC4353259,0x00000000`
158			`.long 0x3FFE0000,0xEE57C16E,0x0D379C0D,0x00000000`
159			`.long 0x3FFE0000,0xF3E10211,0xA87C3779,0x00000000`
160			`.long 0x3FFE0000,0xF919039D,0x758B8D41,0x00000000`
161			`.long 0x3FFE0000,0xFE058B8F,0x64935FB3,0x00000000`
162			`.long 0x3FFF0000,0x8155FB49,0x7B685D04,0x00000000`
163			`.long 0x3FFF0000,0x83889E35,0x49D108E1,0x00000000`
164			`.long 0x3FFF0000,0x859CFA76,0x511D724B,0x00000000`
165			`.long 0x3FFF0000,0x87952ECF,0xFF8131E7,0x00000000`
166			`.long 0x3FFF0000,0x89732FD1,0x9557641B,0x00000000`
167			`.long 0x3FFF0000,0x8B38CAD1,0x01932A35,0x00000000`
168			`.long 0x3FFF0000,0x8CE7A8D8,0x301EE6B5,0x00000000`
169			`.long 0x3FFF0000,0x8F46A39E,0x2EAE5281,0x00000000`
170			`.long 0x3FFF0000,0x922DA7D7,0x91888487,0x00000000`
171			`.long 0x3FFF0000,0x94D19FCB,0xDEDF5241,0x00000000`
172			`.long 0x3FFF0000,0x973AB944,0x19D2A08B,0x00000000`
173			`.long 0x3FFF0000,0x996FF00E,0x08E10B96,0x00000000`
174			`.long 0x3FFF0000,0x9B773F95,0x12321DA7,0x00000000`
175			`.long 0x3FFF0000,0x9D55CC32,0x0F935624,0x00000000`
176			`.long 0x3FFF0000,0x9F100575,0x006CC571,0x00000000`
177			`.long 0x3FFF0000,0xA0A9C290,0xD97CC06C,0x00000000`
178			`.long 0x3FFF0000,0xA22659EB,0xEBC0630A,0x00000000`
179			`.long 0x3FFF0000,0xA388B4AF,0xF6EF0EC9,0x00000000`
180			`.long 0x3FFF0000,0xA4D35F10,0x61D292C4,0x00000000`
181			`.long 0x3FFF0000,0xA60895DC,0xFBE3187E,0x00000000`
182			`.long 0x3FFF0000,0xA72A51DC,0x7367BEAC,0x00000000`
183			`.long 0x3FFF0000,0xA83A5153,0x0956168F,0x00000000`
184			`.long 0x3FFF0000,0xA93A2007,0x7539546E,0x00000000`
185			`.long 0x3FFF0000,0xAA9E7245,0x023B2605,0x00000000`
186			`.long 0x3FFF0000,0xAC4C84BA,0x6FE4D58F,0x00000000`
187			`.long 0x3FFF0000,0xADCE4A4A,0x606B9712,0x00000000`
188			`.long 0x3FFF0000,0xAF2A2DCD,0x8D263C9C,0x00000000`
189			`.long 0x3FFF0000,0xB0656F81,0xF22265C7,0x00000000`
190			`.long 0x3FFF0000,0xB1846515,0x0F71496A,0x00000000`
191			`.long 0x3FFF0000,0xB28AAA15,0x6F9ADA35,0x00000000`
192			`.long 0x3FFF0000,0xB37B44FF,0x3766B895,0x00000000`
193			`.long 0x3FFF0000,0xB458C3DC,0xE9630433,0x00000000`
194			`.long 0x3FFF0000,0xB525529D,0x562246BD,0x00000000`
195			`.long 0x3FFF0000,0xB5E2CCA9,0x5F9D88CC,0x00000000`
196			`.long 0x3FFF0000,0xB692CADA,0x7ACA1ADA,0x00000000`
197			`.long 0x3FFF0000,0xB736AEA7,0xA6925838,0x00000000`
198			`.long 0x3FFF0000,0xB7CFAB28,0x7E9F7B36,0x00000000`
199			`.long 0x3FFF0000,0xB85ECC66,0xCB219835,0x00000000`
200			`.long 0x3FFF0000,0xB8E4FD5A,0x20A593DA,0x00000000`
201			`.long 0x3FFF0000,0xB99F41F6,0x4AFF9BB5,0x00000000`
202			`.long 0x3FFF0000,0xBA7F1E17,0x842BBE7B,0x00000000`
203			`.long 0x3FFF0000,0xBB471285,0x7637E17D,0x00000000`
204			`.long 0x3FFF0000,0xBBFABE8A,0x4788DF6F,0x00000000`
205			`.long 0x3FFF0000,0xBC9D0FAD,0x2B689D79,0x00000000`
206			`.long 0x3FFF0000,0xBD306A39,0x471ECD86,0x00000000`
207			`.long 0x3FFF0000,0xBDB6C731,0x856AF18A,0x00000000`
208			`.long 0x3FFF0000,0xBE31CAC5,0x02E80D70,0x00000000`
209			`.long 0x3FFF0000,0xBEA2D55C,0xE33194E2,0x00000000`
210			`.long 0x3FFF0000,0xBF0B10B7,0xC03128F0,0x00000000`
211			`.long 0x3FFF0000,0xBF6B7A18,0xDACB778D,0x00000000`
212			`.long 0x3FFF0000,0xBFC4EA46,0x63FA18F6,0x00000000`
213			`.long 0x3FFF0000,0xC0181BDE,0x8B89A454,0x00000000`
214			`.long 0x3FFF0000,0xC065B066,0xCFBF6439,0x00000000`
215			`.long 0x3FFF0000,0xC0AE345F,0x56340AE6,0x00000000`
216			`.long 0x3FFF0000,0xC0F22291,0x9CB9E6A7,0x00000000`
217
218			`.set X,FP_SCR1`
219			`.set XDCARE,X+2`
220			`.set XFRAC,X+4`
221			`.set XFRACLO,X+8`
222
223			`.set ATANF,FP_SCR2`
224			`.set ATANFHI,ATANF+4`
225			`.set ATANFLO,ATANF+8`
226
227
228			`\| xref t_frcinx`
229			`\|xref t_extdnrm`
230
231			`.global satand`
232			`satand:`
233			`\|--ENTRY POINT FOR ATAN(X) FOR DENORMALIZED ARGUMENT`
234
235			`bra t_extdnrm`
236
237			`.global satan`
238			`satan:`
239			`\|--ENTRY POINT FOR ATAN(X), HERE X IS FINITE, NON-ZERO, AND NOT NAN'S`
240
241			`fmovex (%a0),%fp0 \| ...LOAD INPUT`
242
243			`movel (%a0),%d0`
244			`movew 4(%a0),%d0`
245			`fmovex %fp0,X(%a6)`
246			`andil #0x7FFFFFFF,%d0`
247
248			`cmpil #0x3FFB8000,%d0 \| ...\|X\| >= 1/16?`
249			`bges ATANOK1`
250			`bra ATANSM`
251
252			`ATANOK1:`
253			`cmpil #0x4002FFFF,%d0 \| ...\|X\| < 16 ?`
254			`bles ATANMAIN`
255			`bra ATANBIG`
256
257
258			`\|--THE MOST LIKELY CASE, \|X\| IN [1/16, 16). WE USE TABLE TECHNIQUE`
259			`\|--THE IDEA IS ATAN(X) = ATAN(F) + ATAN( [X-F] / [1+XF] ).`
260			`\|--SO IF F IS CHOSEN TO BE CLOSE TO X AND ATAN(F) IS STORED IN`
261			`\|--A TABLE, ALL WE NEED IS TO APPROXIMATE ATAN(U) WHERE`
262			`\|--U = (X-F)/(1+XF) IS SMALL (REMEMBER F IS CLOSE TO X). IT IS`
263			`\|--TRUE THAT A DIVIDE IS NOW NEEDED, BUT THE APPROXIMATION FOR`
264			`\|--ATAN(U) IS A VERY SHORT POLYNOMIAL AND THE INDEXING TO`
265			`\|--FETCH F AND SAVING OF REGISTERS CAN BE ALL HIDED UNDER THE`
266			`\|--DIVIDE. IN THE END THIS METHOD IS MUCH FASTER THAN A TRADITIONAL`
267			`\|--ONE. NOTE ALSO THAT THE TRADITIONAL SCHEME THAT APPROXIMATE`
268			`\|--ATAN(X) DIRECTLY WILL NEED TO USE A RATIONAL APPROXIMATION`
269			`\|--(DIVISION NEEDED) ANYWAY BECAUSE A POLYNOMIAL APPROXIMATION`
270			`\|--WILL INVOLVE A VERY LONG POLYNOMIAL.`
271
272			`\|--NOW WE SEE X AS +-2^K * 1.BBBBBBB....B <- 1. + 63 BITS`
273			`\|--WE CHOSE F TO BE +-2^K * 1.BBBB1`
274			`\|--THAT IS IT MATCHES THE EXPONENT AND FIRST 5 BITS OF X, THE`
275			`\|--SIXTH BITS IS SET TO BE 1. SINCE K = -4, -3, ..., 3, THERE`
276			`\|--ARE ONLY 8 TIMES 16 = 2^7 = 128 \|F\|'S. SINCE ATAN(-\|F\|) IS`
277			`\|-- -ATAN(\|F\|), WE NEED TO STORE ONLY ATAN(\|F\|).`
278
279			`ATANMAIN:`
280
281			`movew #0x0000,XDCARE(%a6) \| ...CLEAN UP X JUST IN CASE`
282			`andil #0xF8000000,XFRAC(%a6) \| ...FIRST 5 BITS`
283			`oril #0x04000000,XFRAC(%a6) \| ...SET 6-TH BIT TO 1`
284			`movel #0x00000000,XFRACLO(%a6) \| ...LOCATION OF X IS NOW F`
285
286			`fmovex %fp0,%fp1 \| ...FP1 IS X`
287			`fmulx X(%a6),%fp1 \| ...FP1 IS XF, NOTE THAT XF > 0`
288			`fsubx X(%a6),%fp0 \| ...FP0 IS X-F`
289			`fadds #0x3F800000,%fp1 \| ...FP1 IS 1 + X*F`
290			`fdivx %fp1,%fp0 \| ...FP0 IS U = (X-F)/(1+X*F)`
291
292			`\|--WHILE THE DIVISION IS TAKING ITS TIME, WE FETCH ATAN(\|F\|)`
293			`\|--CREATE ATAN(F) AND STORE IT IN ATANF, AND`
294			`\|--SAVE REGISTERS FP2.`
295
296			`movel %d2,-(%a7) \| ...SAVE d2 TEMPORARILY`
297			`movel %d0,%d2 \| ...THE EXPO AND 16 BITS OF X`
298			`andil #0x00007800,%d0 \| ...4 VARYING BITS OF F'S FRACTION`
299			`andil #0x7FFF0000,%d2 \| ...EXPONENT OF F`
300			`subil #0x3FFB0000,%d2 \| ...K+4`
301			`asrl #1,%d2`
302			`addl %d2,%d0 \| ...THE 7 BITS IDENTIFYING F`
303			`asrl #7,%d0 \| ...INDEX INTO TBL OF ATAN(\|F\|)`
304			`lea ATANTBL,%a1`
305			`addal %d0,%a1 \| ...ADDRESS OF ATAN(\|F\|)`
306			`movel (%a1)+,ATANF(%a6)`
307			`movel (%a1)+,ATANFHI(%a6)`
308			`movel (%a1)+,ATANFLO(%a6) \| ...ATANF IS NOW ATAN(\|F\|)`
309			`movel X(%a6),%d0 \| ...LOAD SIGN AND EXPO. AGAIN`
310			`andil #0x80000000,%d0 \| ...SIGN(F)`
311			`orl %d0,ATANF(%a6) \| ...ATANF IS NOW SIGN(F)*ATAN(\|F\|)`
312			`movel (%a7)+,%d2 \| ...RESTORE d2`
313
314			`\|--THAT'S ALL I HAVE TO DO FOR NOW,`
315			`\|--BUT ALAS, THE DIVIDE IS STILL CRANKING!`
316
317			`\|--U IN FP0, WE ARE NOW READY TO COMPUTE ATAN(U) AS`
318			`\|--U + A1UV(A2 + V(A3 + V)), V = U*U`
319			`\|--THE POLYNOMIAL MAY LOOK STRANGE, BUT IS NEVERTHELESS CORRECT.`
320			`\|--THE NATURAL FORM IS U + UV(A1 + V(A2 + VA3))`
321			`\|--WHAT WE HAVE HERE IS MERELY A1 = A3, A2 = A1/A3, A3 = A2/A3.`
322			`\|--THE REASON FOR THIS REARRANGEMENT IS TO MAKE THE INDEPENDENT`
323			`\|--PARTS A1UV AND (A2 + ... STUFF) MORE LOAD-BALANCED`
324
325
326			`fmovex %fp0,%fp1`
327			`fmulx %fp1,%fp1`
328			`fmoved ATANA3,%fp2`
329			`faddx %fp1,%fp2 \| ...A3+V`
330			`fmulx %fp1,%fp2 \| ...V*(A3+V)`
331			`fmulx %fp0,%fp1 \| ...U*V`
332			`faddd ATANA2,%fp2 \| ...A2+V*(A3+V)`
333			`fmuld ATANA1,%fp1 \| ...A1UV`
334			`fmulx %fp2,%fp1 \| ...A1UV(A2+V(A3+V))`
335
336			`faddx %fp1,%fp0 \| ...ATAN(U), FP1 RELEASED`
337			`fmovel %d1,%FPCR \|restore users exceptions`
338			`faddx ATANF(%a6),%fp0 \| ...ATAN(X)`
339			`bra t_frcinx`
340
341			`ATANBORS:`
342			`\|--\|X\| IS IN d0 IN COMPACT FORM. FP1, d0 SAVED.`
343			`\|--FP0 IS X AND \|X\| <= 1/16 OR \|X\| >= 16.`
344			`cmpil #0x3FFF8000,%d0`
345			`bgt ATANBIG \| ...I.E. \|X\| >= 16`
346
347			`ATANSM:`
348			`\|--\|X\| <= 1/16`
349			`\|--IF \|X\| < 2^(-40), RETURN X AS ANSWER. OTHERWISE, APPROXIMATE`
350			`\|--ATAN(X) BY X + XY(B1+Y(B2+Y(B3+Y(B4+Y(B5+Y*B6)))))`
351			`\|--WHICH IS X + XY( [B1+Z(B3+ZB5)] + [Y(B2+Z(B4+Z*B6)] )`
352			`\|--WHERE Y = XX, AND Z = YY.`
353
354			`cmpil #0x3FD78000,%d0`
355			`blt ATANTINY`
356			`\|--COMPUTE POLYNOMIAL`
357			`fmulx %fp0,%fp0 \| ...FP0 IS Y = X*X`
358
359
360			`movew #0x0000,XDCARE(%a6)`
361
362			`fmovex %fp0,%fp1`
363			`fmulx %fp1,%fp1 \| ...FP1 IS Z = Y*Y`
364
365			`fmoved ATANB6,%fp2`
366			`fmoved ATANB5,%fp3`
367
368			`fmulx %fp1,%fp2 \| ...Z*B6`
369			`fmulx %fp1,%fp3 \| ...Z*B5`
370
371			`faddd ATANB4,%fp2 \| ...B4+Z*B6`
372			`faddd ATANB3,%fp3 \| ...B3+Z*B5`
373
374			`fmulx %fp1,%fp2 \| ...Z(B4+ZB6)`
375			`fmulx %fp3,%fp1 \| ...Z(B3+ZB5)`
376
377			`faddd ATANB2,%fp2 \| ...B2+Z(B4+ZB6)`
378			`faddd ATANB1,%fp1 \| ...B1+Z(B3+ZB5)`
379
380			`fmulx %fp0,%fp2 \| ...Y(B2+Z(B4+Z*B6))`
381			`fmulx X(%a6),%fp0 \| ...X*Y`
382
383			`faddx %fp2,%fp1 \| ...[B1+Z(B3+ZB5)]+[Y(B2+Z(B4+Z*B6))]`
384
385
386			`fmulx %fp1,%fp0 \| ...XY([B1+Z(B3+ZB5)]+[Y(B2+Z(B4+Z*B6))])`
387
388			`fmovel %d1,%FPCR \|restore users exceptions`
389			`faddx X(%a6),%fp0`
390
391			`bra t_frcinx`
392
393			`ATANTINY:`
394			`\|--\|X\| < 2^(-40), ATAN(X) = X`
395			`movew #0x0000,XDCARE(%a6)`
396
397			`fmovel %d1,%FPCR \|restore users exceptions`
398			`fmovex X(%a6),%fp0 \|last inst - possible exception set`
399
400			`bra t_frcinx`
401
402			`ATANBIG:`
403			`\|--IF \|X\| > 2^(100), RETURN SIGN(X)*(PI/2 - TINY). OTHERWISE,`
404			`\|--RETURN SIGN(X)*PI/2 + ATAN(-1/X).`
405			`cmpil #0x40638000,%d0`
406			`bgt ATANHUGE`
407
408			`\|--APPROXIMATE ATAN(-1/X) BY`
409			`\|--X'+X'Y(C1+Y(C2+Y(C3+Y(C4+YC5)))), X' = -1/X, Y = X'*X'`
410			`\|--THIS CAN BE RE-WRITTEN AS`
411			`\|--X'+X'Y( [C1+Z(C3+ZC5)] + [Y(C2+ZC4)] ), Z = Y*Y.`
412
413			`fmoves #0xBF800000,%fp1 \| ...LOAD -1`
414			`fdivx %fp0,%fp1 \| ...FP1 IS -1/X`
415
416
417			`\|--DIVIDE IS STILL CRANKING`
418
419			`fmovex %fp1,%fp0 \| ...FP0 IS X'`
420			`fmulx %fp0,%fp0 \| ...FP0 IS Y = X'*X'`
421			`fmovex %fp1,X(%a6) \| ...X IS REALLY X'`
422
423			`fmovex %fp0,%fp1`
424			`fmulx %fp1,%fp1 \| ...FP1 IS Z = Y*Y`
425
426			`fmoved ATANC5,%fp3`
427			`fmoved ATANC4,%fp2`
428
429			`fmulx %fp1,%fp3 \| ...Z*C5`
430			`fmulx %fp1,%fp2 \| ...Z*B4`
431
432			`faddd ATANC3,%fp3 \| ...C3+Z*C5`
433			`faddd ATANC2,%fp2 \| ...C2+Z*C4`
434
435			`fmulx %fp3,%fp1 \| ...Z(C3+ZC5), FP3 RELEASED`
436			`fmulx %fp0,%fp2 \| ...Y(C2+ZC4)`
437
438			`faddd ATANC1,%fp1 \| ...C1+Z(C3+ZC5)`
439			`fmulx X(%a6),%fp0 \| ...X'*Y`
440
441			`faddx %fp2,%fp1 \| ...[Y(C2+ZC4)]+[C1+Z(C3+ZC5)]`
442
443
444			`fmulx %fp1,%fp0 \| ...X'Y([B1+Z(B3+ZB5)]`
445			`\| ... +[Y(B2+Z(B4+Z*B6))])`
446			`faddx X(%a6),%fp0`
447
448			`fmovel %d1,%FPCR \|restore users exceptions`
449
450			`btstb #7,(%a0)`
451			`beqs pos_big`
452
453			`neg_big:`
454			`faddx NPIBY2,%fp0`
455			`bra t_frcinx`
456
457			`pos_big:`
458			`faddx PPIBY2,%fp0`
459			`bra t_frcinx`
460
461			`ATANHUGE:`
462			`\|--RETURN SIGN(X)(PIBY2 - TINY) = SIGN(X)PIBY2 - SIGN(X)*TINY`
463			`btstb #7,(%a0)`
464			`beqs pos_huge`
465
466			`neg_huge:`
467			`fmovex NPIBY2,%fp0`
468			`fmovel %d1,%fpcr`
469			`fsubx NTINY,%fp0`
470			`bra t_frcinx`
471
472			`pos_huge:`
473			`fmovex PPIBY2,%fp0`
474			`fmovel %d1,%fpcr`
475			`fsubx PTINY,%fp0`
476			`bra t_frcinx`
477
478			`\|end`