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|
|       stwotox.sa 3.1 12/10/90
|
|       stwotox  --- 2**X
|       stwotoxd --- 2**X for denormalized X
|       stentox  --- 10**X
|       stentoxd --- 10**X for denormalized X
|
|       Input: Double-extended number X in location pointed to
|               by address register a0.
|
|       Output: The function values are returned in 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 stwotox takes approximately 190 cycles and the
|               program stentox takes approximately 200 cycles.
|
|       Algorithm:
|
|       twotox
|       1. If |X| > 16480, go to ExpBig.
|
|       2. If |X| < 2**(-70), go to ExpSm.
|
|       3. Decompose X as X = N/64 + r where |r| <= 1/128. Furthermore
|               decompose N as
|                N = 64(M + M') + j,  j = 0,1,2,...,63.
|
|       4. Overwrite r := r * log2. Then
|               2**X = 2**(M') * 2**(M) * 2**(j/64) * exp(r).
|               Go to expr to compute that expression.
|
|       tentox
|       1. If |X| > 16480*log_10(2) (base 10 log of 2), go to ExpBig.
|
|       2. If |X| < 2**(-70), go to ExpSm.
|
|       3. Set y := X*log_2(10)*64 (base 2 log of 10). Set
|               N := round-to-int(y). Decompose N as
|                N = 64(M + M') + j,  j = 0,1,2,...,63.
|
|       4. Define r as
|               r := ((X - N*L1)-N*L2) * L10
|               where L1, L2 are the leading and trailing parts of log_10(2)/64
|               and L10 is the natural log of 10. Then
|               10**X = 2**(M') * 2**(M) * 2**(j/64) * exp(r).
|               Go to expr to compute that expression.
|
|       expr
|       1. Fetch 2**(j/64) from table as Fact1 and Fact2.
|
|       2. Overwrite Fact1 and Fact2 by
|               Fact1 := 2**(M) * Fact1
|               Fact2 := 2**(M) * Fact2
|               Thus Fact1 + Fact2 = 2**(M) * 2**(j/64).
|
|       3. Calculate P where 1 + P approximates exp(r):
|               P = r + r*r*(A1+r*(A2+...+r*A5)).
|
|       4. Let AdjFact := 2**(M'). Return
|               AdjFact * ( Fact1 + ((Fact1*P) + Fact2) ).
|               Exit.
|
|       ExpBig
|       1. Generate overflow by Huge * Huge if X > 0; otherwise, generate
|               underflow by Tiny * Tiny.
|
|       ExpSm
|       1. Return 1 + X.
|

|               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.

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

        |section        8

        .include "fpsp.h"

BOUNDS1:        .long 0x3FB98000,0x400D80C0 | ... 2^(-70),16480
BOUNDS2:        .long 0x3FB98000,0x400B9B07 | ... 2^(-70),16480 LOG2/LOG10

L2TEN64:        .long 0x406A934F,0x0979A371 | ... 64LOG10/LOG2
L10TWO1:        .long 0x3F734413,0x509F8000 | ... LOG2/64LOG10

L10TWO2:        .long 0xBFCD0000,0xC0219DC1,0xDA994FD2,0x00000000

LOG10:  .long 0x40000000,0x935D8DDD,0xAAA8AC17,0x00000000

LOG2:   .long 0x3FFE0000,0xB17217F7,0xD1CF79AC,0x00000000

EXPA5:  .long 0x3F56C16D,0x6F7BD0B2
EXPA4:  .long 0x3F811112,0x302C712C
EXPA3:  .long 0x3FA55555,0x55554CC1
EXPA2:  .long 0x3FC55555,0x55554A54
EXPA1:  .long 0x3FE00000,0x00000000,0x00000000,0x00000000

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

EXPTBL:
        .long  0x3FFF0000,0x80000000,0x00000000,0x3F738000
        .long  0x3FFF0000,0x8164D1F3,0xBC030773,0x3FBEF7CA
        .long  0x3FFF0000,0x82CD8698,0xAC2BA1D7,0x3FBDF8A9
        .long  0x3FFF0000,0x843A28C3,0xACDE4046,0x3FBCD7C9
        .long  0x3FFF0000,0x85AAC367,0xCC487B15,0xBFBDE8DA
        .long  0x3FFF0000,0x871F6196,0x9E8D1010,0x3FBDE85C
        .long  0x3FFF0000,0x88980E80,0x92DA8527,0x3FBEBBF1
        .long  0x3FFF0000,0x8A14D575,0x496EFD9A,0x3FBB80CA
        .long  0x3FFF0000,0x8B95C1E3,0xEA8BD6E7,0xBFBA8373
        .long  0x3FFF0000,0x8D1ADF5B,0x7E5BA9E6,0xBFBE9670
        .long  0x3FFF0000,0x8EA4398B,0x45CD53C0,0x3FBDB700
        .long  0x3FFF0000,0x9031DC43,0x1466B1DC,0x3FBEEEB0
        .long  0x3FFF0000,0x91C3D373,0xAB11C336,0x3FBBFD6D
        .long  0x3FFF0000,0x935A2B2F,0x13E6E92C,0xBFBDB319
        .long  0x3FFF0000,0x94F4EFA8,0xFEF70961,0x3FBDBA2B
        .long  0x3FFF0000,0x96942D37,0x20185A00,0x3FBE91D5
        .long  0x3FFF0000,0x9837F051,0x8DB8A96F,0x3FBE8D5A
        .long  0x3FFF0000,0x99E04593,0x20B7FA65,0xBFBCDE7B
        .long  0x3FFF0000,0x9B8D39B9,0xD54E5539,0xBFBEBAAF
        .long  0x3FFF0000,0x9D3ED9A7,0x2CFFB751,0xBFBD86DA
        .long  0x3FFF0000,0x9EF53260,0x91A111AE,0xBFBEBEDD
        .long  0x3FFF0000,0xA0B0510F,0xB9714FC2,0x3FBCC96E
        .long  0x3FFF0000,0xA2704303,0x0C496819,0xBFBEC90B
        .long  0x3FFF0000,0xA43515AE,0x09E6809E,0x3FBBD1DB
        .long  0x3FFF0000,0xA5FED6A9,0xB15138EA,0x3FBCE5EB
        .long  0x3FFF0000,0xA7CD93B4,0xE965356A,0xBFBEC274
        .long  0x3FFF0000,0xA9A15AB4,0xEA7C0EF8,0x3FBEA83C
        .long  0x3FFF0000,0xAB7A39B5,0xA93ED337,0x3FBECB00
        .long  0x3FFF0000,0xAD583EEA,0x42A14AC6,0x3FBE9301
        .long  0x3FFF0000,0xAF3B78AD,0x690A4375,0xBFBD8367
        .long  0x3FFF0000,0xB123F581,0xD2AC2590,0xBFBEF05F
        .long  0x3FFF0000,0xB311C412,0xA9112489,0x3FBDFB3C
        .long  0x3FFF0000,0xB504F333,0xF9DE6484,0x3FBEB2FB
        .long  0x3FFF0000,0xB6FD91E3,0x28D17791,0x3FBAE2CB
        .long  0x3FFF0000,0xB8FBAF47,0x62FB9EE9,0x3FBCDC3C
        .long  0x3FFF0000,0xBAFF5AB2,0x133E45FB,0x3FBEE9AA
        .long  0x3FFF0000,0xBD08A39F,0x580C36BF,0xBFBEAEFD
        .long  0x3FFF0000,0xBF1799B6,0x7A731083,0xBFBCBF51
        .long  0x3FFF0000,0xC12C4CCA,0x66709456,0x3FBEF88A
        .long  0x3FFF0000,0xC346CCDA,0x24976407,0x3FBD83B2
        .long  0x3FFF0000,0xC5672A11,0x5506DADD,0x3FBDF8AB
        .long  0x3FFF0000,0xC78D74C8,0xABB9B15D,0xBFBDFB17
        .long  0x3FFF0000,0xC9B9BD86,0x6E2F27A3,0xBFBEFE3C
        .long  0x3FFF0000,0xCBEC14FE,0xF2727C5D,0xBFBBB6F8
        .long  0x3FFF0000,0xCE248C15,0x1F8480E4,0xBFBCEE53
        .long  0x3FFF0000,0xD06333DA,0xEF2B2595,0xBFBDA4AE
        .long  0x3FFF0000,0xD2A81D91,0xF12AE45A,0x3FBC9124
        .long  0x3FFF0000,0xD4F35AAB,0xCFEDFA1F,0x3FBEB243
        .long  0x3FFF0000,0xD744FCCA,0xD69D6AF4,0x3FBDE69A
        .long  0x3FFF0000,0xD99D15C2,0x78AFD7B6,0xBFB8BC61
        .long  0x3FFF0000,0xDBFBB797,0xDAF23755,0x3FBDF610
        .long  0x3FFF0000,0xDE60F482,0x5E0E9124,0xBFBD8BE1
        .long  0x3FFF0000,0xE0CCDEEC,0x2A94E111,0x3FBACB12
        .long  0x3FFF0000,0xE33F8972,0xBE8A5A51,0x3FBB9BFE
        .long  0x3FFF0000,0xE5B906E7,0x7C8348A8,0x3FBCF2F4
        .long  0x3FFF0000,0xE8396A50,0x3C4BDC68,0x3FBEF22F
        .long  0x3FFF0000,0xEAC0C6E7,0xDD24392F,0xBFBDBF4A
        .long  0x3FFF0000,0xED4F301E,0xD9942B84,0x3FBEC01A
        .long  0x3FFF0000,0xEFE4B99B,0xDCDAF5CB,0x3FBE8CAC
        .long  0x3FFF0000,0xF281773C,0x59FFB13A,0xBFBCBB3F
        .long  0x3FFF0000,0xF5257D15,0x2486CC2C,0x3FBEF73A
        .long  0x3FFF0000,0xF7D0DF73,0x0AD13BB9,0xBFB8B795
        .long  0x3FFF0000,0xFA83B2DB,0x722A033A,0x3FBEF84B
        .long  0x3FFF0000,0xFD3E0C0C,0xF486C175,0xBFBEF581

        .set    N,L_SCR1

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

        .set    ADJFACT,FP_SCR2

        .set    FACT1,FP_SCR3
        .set    FACT1HI,FACT1+4
        .set    FACT1LOW,FACT1+8

        .set    FACT2,FP_SCR4
        .set    FACT2HI,FACT2+4
        .set    FACT2LOW,FACT2+8

        | xref  t_unfl
        |xref   t_ovfl
        |xref   t_frcinx

        .global stwotoxd
stwotoxd:
|--ENTRY POINT FOR 2**(X) FOR DENORMALIZED ARGUMENT

        fmovel          %d1,%fpcr               | ...set user's rounding mode/precision
        fmoves          #0x3F800000,%fp0  | ...RETURN 1 + X
        movel           (%a0),%d0
        orl             #0x00800001,%d0
        fadds           %d0,%fp0
        bra             t_frcinx

        .global stwotox
stwotox:
|--ENTRY POINT FOR 2**(X), HERE X IS FINITE, NON-ZERO, AND NOT NAN'S
        fmovemx (%a0),%fp0-%fp0 | ...LOAD INPUT, do not set cc's

        movel           (%a0),%d0
        movew           4(%a0),%d0
        fmovex          %fp0,X(%a6)
        andil           #0x7FFFFFFF,%d0

        cmpil           #0x3FB98000,%d0         | ...|X| >= 2**(-70)?
        bges            TWOOK1
        bra             EXPBORS

TWOOK1:
        cmpil           #0x400D80C0,%d0         | ...|X| > 16480?
        bles            TWOMAIN
        bra             EXPBORS
        

TWOMAIN:
|--USUAL CASE, 2^(-70) <= |X| <= 16480

        fmovex          %fp0,%fp1
        fmuls           #0x42800000,%fp1  | ...64 * X
        
        fmovel          %fp1,N(%a6)             | ...N = ROUND-TO-INT(64 X)
        movel           %d2,-(%sp)
        lea             EXPTBL,%a1      | ...LOAD ADDRESS OF TABLE OF 2^(J/64)
        fmovel          N(%a6),%fp1             | ...N --> FLOATING FMT
        movel           N(%a6),%d0
        movel           %d0,%d2
        andil           #0x3F,%d0               | ...D0 IS J
        asll            #4,%d0          | ...DISPLACEMENT FOR 2^(J/64)
        addal           %d0,%a1         | ...ADDRESS FOR 2^(J/64)
        asrl            #6,%d2          | ...d2 IS L, N = 64L + J
        movel           %d2,%d0
        asrl            #1,%d0          | ...D0 IS M
        subl            %d0,%d2         | ...d2 IS M', N = 64(M+M') + J
        addil           #0x3FFF,%d2
        movew           %d2,ADJFACT(%a6)        | ...ADJFACT IS 2^(M')
        movel           (%sp)+,%d2
|--SUMMARY: a1 IS ADDRESS FOR THE LEADING PORTION OF 2^(J/64),
|--D0 IS M WHERE N = 64(M+M') + J. NOTE THAT |M| <= 16140 BY DESIGN.
|--ADJFACT = 2^(M').
|--REGISTERS SAVED SO FAR ARE (IN ORDER) FPCR, D0, FP1, a1, AND FP2.

        fmuls           #0x3C800000,%fp1  | ...(1/64)*N
        movel           (%a1)+,FACT1(%a6)
        movel           (%a1)+,FACT1HI(%a6)
        movel           (%a1)+,FACT1LOW(%a6)
        movew           (%a1)+,FACT2(%a6)
        clrw            FACT2+2(%a6)

        fsubx           %fp1,%fp0               | ...X - (1/64)*INT(64 X)

        movew           (%a1)+,FACT2HI(%a6)
        clrw            FACT2HI+2(%a6)
        clrl            FACT2LOW(%a6)
        addw            %d0,FACT1(%a6)
        
        fmulx           LOG2,%fp0       | ...FP0 IS R
        addw            %d0,FACT2(%a6)

        bra             expr

EXPBORS:
|--FPCR, D0 SAVED
        cmpil           #0x3FFF8000,%d0
        bgts            EXPBIG

EXPSM:
|--|X| IS SMALL, RETURN 1 + X

        fmovel          %d1,%FPCR               |restore users exceptions
        fadds           #0x3F800000,%fp0  | ...RETURN 1 + X

        bra             t_frcinx

EXPBIG:
|--|X| IS LARGE, GENERATE OVERFLOW IF X > 0; ELSE GENERATE UNDERFLOW
|--REGISTERS SAVE SO FAR ARE FPCR AND  D0
        movel           X(%a6),%d0
        cmpil           #0,%d0
        blts            EXPNEG

        bclrb           #7,(%a0)                |t_ovfl expects positive value
        bra             t_ovfl

EXPNEG:
        bclrb           #7,(%a0)                |t_unfl expects positive value
        bra             t_unfl

        .global stentoxd
stentoxd:
|--ENTRY POINT FOR 10**(X) FOR DENORMALIZED ARGUMENT

        fmovel          %d1,%fpcr               | ...set user's rounding mode/precision
        fmoves          #0x3F800000,%fp0  | ...RETURN 1 + X
        movel           (%a0),%d0
        orl             #0x00800001,%d0
        fadds           %d0,%fp0
        bra             t_frcinx

        .global stentox
stentox:
|--ENTRY POINT FOR 10**(X), HERE X IS FINITE, NON-ZERO, AND NOT NAN'S
        fmovemx (%a0),%fp0-%fp0 | ...LOAD INPUT, do not set cc's

        movel           (%a0),%d0
        movew           4(%a0),%d0
        fmovex          %fp0,X(%a6)
        andil           #0x7FFFFFFF,%d0

        cmpil           #0x3FB98000,%d0         | ...|X| >= 2**(-70)?
        bges            TENOK1
        bra             EXPBORS

TENOK1:
        cmpil           #0x400B9B07,%d0         | ...|X| <= 16480*log2/log10 ?
        bles            TENMAIN
        bra             EXPBORS

TENMAIN:
|--USUAL CASE, 2^(-70) <= |X| <= 16480 LOG 2 / LOG 10

        fmovex          %fp0,%fp1
        fmuld           L2TEN64,%fp1    | ...X*64*LOG10/LOG2
        
        fmovel          %fp1,N(%a6)             | ...N=INT(X*64*LOG10/LOG2)
        movel           %d2,-(%sp)
        lea             EXPTBL,%a1      | ...LOAD ADDRESS OF TABLE OF 2^(J/64)
        fmovel          N(%a6),%fp1             | ...N --> FLOATING FMT
        movel           N(%a6),%d0
        movel           %d0,%d2
        andil           #0x3F,%d0               | ...D0 IS J
        asll            #4,%d0          | ...DISPLACEMENT FOR 2^(J/64)
        addal           %d0,%a1         | ...ADDRESS FOR 2^(J/64)
        asrl            #6,%d2          | ...d2 IS L, N = 64L + J
        movel           %d2,%d0
        asrl            #1,%d0          | ...D0 IS M
        subl            %d0,%d2         | ...d2 IS M', N = 64(M+M') + J
        addil           #0x3FFF,%d2
        movew           %d2,ADJFACT(%a6)        | ...ADJFACT IS 2^(M')
        movel           (%sp)+,%d2

|--SUMMARY: a1 IS ADDRESS FOR THE LEADING PORTION OF 2^(J/64),
|--D0 IS M WHERE N = 64(M+M') + J. NOTE THAT |M| <= 16140 BY DESIGN.
|--ADJFACT = 2^(M').
|--REGISTERS SAVED SO FAR ARE (IN ORDER) FPCR, D0, FP1, a1, AND FP2.

        fmovex          %fp1,%fp2

        fmuld           L10TWO1,%fp1    | ...N*(LOG2/64LOG10)_LEAD
        movel           (%a1)+,FACT1(%a6)

        fmulx           L10TWO2,%fp2    | ...N*(LOG2/64LOG10)_TRAIL

        movel           (%a1)+,FACT1HI(%a6)
        movel           (%a1)+,FACT1LOW(%a6)
        fsubx           %fp1,%fp0               | ...X - N L_LEAD
        movew           (%a1)+,FACT2(%a6)

        fsubx           %fp2,%fp0               | ...X - N L_TRAIL

        clrw            FACT2+2(%a6)
        movew           (%a1)+,FACT2HI(%a6)
        clrw            FACT2HI+2(%a6)
        clrl            FACT2LOW(%a6)

        fmulx           LOG10,%fp0      | ...FP0 IS R
        
        addw            %d0,FACT1(%a6)
        addw            %d0,FACT2(%a6)

expr:
|--FPCR, FP2, FP3 ARE SAVED IN ORDER AS SHOWN.
|--ADJFACT CONTAINS 2**(M'), FACT1 + FACT2 = 2**(M) * 2**(J/64).
|--FP0 IS R. THE FOLLOWING CODE COMPUTES
|--     2**(M'+M) * 2**(J/64) * EXP(R)

        fmovex          %fp0,%fp1
        fmulx           %fp1,%fp1               | ...FP1 IS S = R*R

        fmoved          EXPA5,%fp2      | ...FP2 IS A5
        fmoved          EXPA4,%fp3      | ...FP3 IS A4

        fmulx           %fp1,%fp2               | ...FP2 IS S*A5
        fmulx           %fp1,%fp3               | ...FP3 IS S*A4

        faddd           EXPA3,%fp2      | ...FP2 IS A3+S*A5
        faddd           EXPA2,%fp3      | ...FP3 IS A2+S*A4

        fmulx           %fp1,%fp2               | ...FP2 IS S*(A3+S*A5)
        fmulx           %fp1,%fp3               | ...FP3 IS S*(A2+S*A4)

        faddd           EXPA1,%fp2      | ...FP2 IS A1+S*(A3+S*A5)
        fmulx           %fp0,%fp3               | ...FP3 IS R*S*(A2+S*A4)

        fmulx           %fp1,%fp2               | ...FP2 IS S*(A1+S*(A3+S*A5))
        faddx           %fp3,%fp0               | ...FP0 IS R+R*S*(A2+S*A4)
        
        faddx           %fp2,%fp0               | ...FP0 IS EXP(R) - 1
        

|--FINAL RECONSTRUCTION PROCESS
|--EXP(X) = 2^M*2^(J/64) + 2^M*2^(J/64)*(EXP(R)-1)  -  (1 OR 0)

        fmulx           FACT1(%a6),%fp0
        faddx           FACT2(%a6),%fp0
        faddx           FACT1(%a6),%fp0

        fmovel          %d1,%FPCR               |restore users exceptions
        clrw            ADJFACT+2(%a6)
        movel           #0x80000000,ADJFACT+4(%a6)
        clrl            ADJFACT+8(%a6)
        fmulx           ADJFACT(%a6),%fp0       | ...FINAL ADJUSTMENT

        bra             t_frcinx

        |end