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|

|       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

|

|       THIS IS UNPUBLISHED PROPRIETARY SOURCE CODE OF MOTOROLA

|       The copyright notice above does not evidence any

|       actual or intended publication of such source code.

|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

        addl    ADJK(%a6),%d0   | ...ADJUST K, ORIGINAL INPUT MAY BE  DENORM.

        lea     LOGTBL,%a0      | ...BASE ADDRESS OF 1/F AND LOG(F)

        fmovel  %d0,%fp1                | ...CONVERT K TO FLOATING-POINT FORMAT

|--WHILE THE CONVERSION IS GOING ON, WE GET F AND ADDRESS OF 1/F

        movel   #0x3FFF0000,X(%a6)      | ...X IS NOW Y, I.E. 2^(-K)*X

        movel   XFRAC(%a6),FFRAC(%a6)

        andil   #0xFE000000,FFRAC(%a6) | ...FIRST 7 BITS OF Y

        oril    #0x01000000,FFRAC(%a6) | ...GET F: ATTACH A 1 AT THE EIGHTH BIT

        movel   FFRAC(%a6),%d0  | ...READY TO GET ADDRESS OF 1/F

        andil   #0x7E000000,%d0

        asrl    #8,%d0

        asrl    #8,%d0

        asrl    #4,%d0          | ...SHIFTED 20, D0 IS THE DISPLACEMENT

        addal   %d0,%a0         | ...A0 IS THE ADDRESS FOR 1/F

        fmovex  X(%a6),%fp0

        movel   #0x3fff0000,F(%a6)

        clrl    F+8(%a6)

        fsubx   F(%a6),%fp0             | ...Y-F

        fmovemx %fp2-%fp2/%fp3,-(%sp)   | ...SAVE FP2 WHILE FP0 IS NOT READY

|--SUMMARY: FP0 IS Y-F, A0 IS ADDRESS OF 1/F, FP1 IS K

|--REGISTERS SAVED: FPCR, FP1, FP2

LP1CONT1:

|--AN RE-ENTRY POINT FOR LOGNP1

        fmulx   (%a0),%fp0      | ...FP0 IS U = (Y-F)/F

        fmulx   LOGOF2,%fp1     | ...GET K*LOG2 WHILE FP0 IS NOT READY

        fmovex  %fp0,%fp2

        fmulx   %fp2,%fp2               | ...FP2 IS V=U*U

        fmovex  %fp1,KLOG2(%a6) | ...PUT K*LOG2 IN MEMORY, FREE FP1

|--LOG(1+U) IS APPROXIMATED BY

|--U + V*(A1+U*(A2+U*(A3+U*(A4+U*(A5+U*A6))))) WHICH IS

|--[U + V*(A1+V*(A3+V*A5))]  +  [U*V*(A2+V*(A4+V*A6))]

        fmovex  %fp2,%fp3

        fmovex  %fp2,%fp1

        fmuld   LOGA6,%fp1      | ...V*A6

        fmuld   LOGA5,%fp2      | ...V*A5

        faddd   LOGA4,%fp1      | ...A4+V*A6

        faddd   LOGA3,%fp2      | ...A3+V*A5

        fmulx   %fp3,%fp1               | ...V*(A4+V*A6)

        fmulx   %fp3,%fp2               | ...V*(A3+V*A5)

        faddd   LOGA2,%fp1      | ...A2+V*(A4+V*A6)

        faddd   LOGA1,%fp2      | ...A1+V*(A3+V*A5)

        fmulx   %fp3,%fp1               | ...V*(A2+V*(A4+V*A6))

        addal   #16,%a0         | ...ADDRESS OF LOG(F)

        fmulx   %fp3,%fp2               | ...V*(A1+V*(A3+V*A5)), FP3 RELEASED

        fmulx   %fp0,%fp1               | ...U*V*(A2+V*(A4+V*A6))

        faddx   %fp2,%fp0               | ...U+V*(A1+V*(A3+V*A5)), FP2 RELEASED

        faddx   (%a0),%fp1      | ...LOG(F)+U*V*(A2+V*(A4+V*A6))

        fmovemx  (%sp)+,%fp2-%fp2/%fp3  | ...RESTORE FP2

        faddx   %fp1,%fp0               | ...FP0 IS LOG(F) + LOG(1+U)

        fmovel  %d1,%fpcr

        faddx   KLOG2(%a6),%fp0 | ...FINAL ADD

        bra     t_frcinx

LOGNEAR1:

|--REGISTERS SAVED: FPCR, FP1. FP0 CONTAINS THE INPUT.

        fmovex  %fp0,%fp1

        fsubs   one,%fp1                | ...FP1 IS X-1

        fadds   one,%fp0                | ...FP0 IS X+1

        faddx   %fp1,%fp1               | ...FP1 IS 2(X-1)

|--LOG(X) = LOG(1+U/2)-LOG(1-U/2) WHICH IS AN ODD POLYNOMIAL

|--IN U, U = 2(X-1)/(X+1) = FP1/FP0

LP1CONT2:

|--THIS IS AN RE-ENTRY POINT FOR LOGNP1

        fdivx   %fp0,%fp1               | ...FP1 IS U

        fmovemx %fp2-%fp2/%fp3,-(%sp)    | ...SAVE FP2

|--REGISTERS SAVED ARE NOW FPCR,FP1,FP2,FP3

|--LET V=U*U, W=V*V, CALCULATE

|--U + U*V*(B1 + V*(B2 + V*(B3 + V*(B4 + V*B5)))) BY

|--U + U*V*(  [B1 + W*(B3 + W*B5)]  +  [V*(B2 + W*B4)]  )

        fmovex  %fp1,%fp0

        fmulx   %fp0,%fp0       | ...FP0 IS V

        fmovex  %fp1,SAVEU(%a6) | ...STORE U IN MEMORY, FREE FP1

        fmovex  %fp0,%fp1

        fmulx   %fp1,%fp1       | ...FP1 IS W

        fmoved  LOGB5,%fp3

        fmoved  LOGB4,%fp2

        fmulx   %fp1,%fp3       | ...W*B5

        fmulx   %fp1,%fp2       | ...W*B4

        faddd   LOGB3,%fp3 | ...B3+W*B5

        faddd   LOGB2,%fp2 | ...B2+W*B4

        fmulx   %fp3,%fp1       | ...W*(B3+W*B5), FP3 RELEASED

        fmulx   %fp0,%fp2       | ...V*(B2+W*B4)

        faddd   LOGB1,%fp1 | ...B1+W*(B3+W*B5)

        fmulx   SAVEU(%a6),%fp0 | ...FP0 IS U*V

        faddx   %fp2,%fp1       | ...B1+W*(B3+W*B5) + V*(B2+W*B4), FP2 RELEASED

        fmovemx (%sp)+,%fp2-%fp2/%fp3 | ...FP2 RESTORED

        fmulx   %fp1,%fp0       | ...U*V*( [B1+W*(B3+W*B5)] + [V*(B2+W*B4)] )

        fmovel  %d1,%fpcr

        faddx   SAVEU(%a6),%fp0

        bra     t_frcinx

        rts

LOGNEG:

|--REGISTERS SAVED FPCR. LOG(-VE) IS INVALID

        bra     t_operr

        .global slognp1d

slognp1d:

|--ENTRY POINT FOR LOG(1+Z) FOR DENORMALIZED INPUT

| Simply return the denorm

        bra     t_extdnrm

        .global slognp1

slognp1:

|--ENTRY POINT FOR LOG(1+X) FOR X FINITE, NON-ZERO, NOT NAN'S

        fmovex  (%a0),%fp0      | ...LOAD INPUT

        fabsx   %fp0            |test magnitude

        fcmpx   LTHOLD,%fp0     |compare with min threshold

        fbgt    LP1REAL         |if greater, continue

        fmovel  #0,%fpsr                |clr N flag from compare

        fmovel  %d1,%fpcr

        fmovex  (%a0),%fp0      |return signed argument

        bra     t_frcinx

LP1REAL:

        fmovex  (%a0),%fp0      | ...LOAD INPUT

        movel   #0x00000000,ADJK(%a6)

        fmovex  %fp0,%fp1       | ...FP1 IS INPUT Z

        fadds   one,%fp0        | ...X := ROUND(1+Z)

        fmovex  %fp0,X(%a6)

        movew   XFRAC(%a6),XDCARE(%a6)

        movel   X(%a6),%d0

        cmpil   #0,%d0

        ble     LP1NEG0 | ...LOG OF ZERO OR -VE

        cmp2l   BOUNDS2,%d0

        bcs     LOGMAIN | ...BOUNDS2 IS [1/2,3/2]

|--IF 1+Z > 3/2 OR 1+Z < 1/2, THEN X, WHICH IS ROUNDING 1+Z,

|--CONTAINS AT LEAST 63 BITS OF INFORMATION OF Z. IN THAT CASE,

|--SIMPLY INVOKE LOG(X) FOR LOG(1+Z).

LP1NEAR1:

|--NEXT SEE IF EXP(-1/16) < X < EXP(1/16)

        cmp2l   BOUNDS1,%d0

        bcss    LP1CARE

LP1ONE16:

|--EXP(-1/16) < X < EXP(1/16). LOG(1+Z) = LOG(1+U/2) - LOG(1-U/2)

|--WHERE U = 2Z/(2+Z) = 2Z/(1+X).

        faddx   %fp1,%fp1       | ...FP1 IS 2Z

        fadds   one,%fp0        | ...FP0 IS 1+X

|--U = FP1/FP0

        bra     LP1CONT2

LP1CARE:

|--HERE WE USE THE USUAL TABLE DRIVEN APPROACH. CARE HAS TO BE

|--TAKEN BECAUSE 1+Z CAN HAVE 67 BITS OF INFORMATION AND WE MUST

|--PRESERVE ALL THE INFORMATION. BECAUSE 1+Z IS IN [1/2,3/2],

|--THERE ARE ONLY TWO CASES.

|--CASE 1: 1+Z < 1, THEN K = -1 AND Y-F = (2-F) + 2Z

|--CASE 2: 1+Z > 1, THEN K = 0  AND Y-F = (1-F) + Z

|--ON RETURNING TO LP1CONT1, WE MUST HAVE K IN FP1, ADDRESS OF

|--(1/F) IN A0, Y-F IN FP0, AND FP2 SAVED.

        movel   XFRAC(%a6),FFRAC(%a6)

        andil   #0xFE000000,FFRAC(%a6)

        oril    #0x01000000,FFRAC(%a6)  | ...F OBTAINED

        cmpil   #0x3FFF8000,%d0 | ...SEE IF 1+Z > 1

        bges    KISZERO

KISNEG1:

        fmoves  TWO,%fp0

        movel   #0x3fff0000,F(%a6)

        clrl    F+8(%a6)

        fsubx   F(%a6),%fp0     | ...2-F

        movel   FFRAC(%a6),%d0

        andil   #0x7E000000,%d0

        asrl    #8,%d0

        asrl    #8,%d0

        asrl    #4,%d0          | ...D0 CONTAINS DISPLACEMENT FOR 1/F

        faddx   %fp1,%fp1               | ...GET 2Z

        fmovemx %fp2-%fp2/%fp3,-(%sp)   | ...SAVE FP2

        faddx   %fp1,%fp0               | ...FP0 IS Y-F = (2-F)+2Z

        lea     LOGTBL,%a0      | ...A0 IS ADDRESS OF 1/F

        addal   %d0,%a0

        fmoves  negone,%fp1     | ...FP1 IS K = -1

        bra     LP1CONT1

KISZERO:

        fmoves  one,%fp0

        movel   #0x3fff0000,F(%a6)

        clrl    F+8(%a6)

        fsubx   F(%a6),%fp0             | ...1-F

        movel   FFRAC(%a6),%d0

        andil   #0x7E000000,%d0

        asrl    #8,%d0

        asrl    #8,%d0

        asrl    #4,%d0

        faddx   %fp1,%fp0               | ...FP0 IS Y-F

        fmovemx %fp2-%fp2/%fp3,-(%sp)   | ...FP2 SAVED

        lea     LOGTBL,%a0

        addal   %d0,%a0         | ...A0 IS ADDRESS OF 1/F

        fmoves  zero,%fp1       | ...FP1 IS K = 0

        bra     LP1CONT1

LP1NEG0:

|--FPCR SAVED. D0 IS X IN COMPACT FORM.

        cmpil   #0,%d0

        blts    LP1NEG

LP1ZERO:

        fmoves  negone,%fp0

        fmovel  %d1,%fpcr

        bra t_dz

LP1NEG:

        fmoves  zero,%fp0

        fmovel  %d1,%fpcr

        bra     t_operr

        |end