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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| THIS IS UNPUBLISHED PROPRIETARY SOURCE CODE OF MOTOROLA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
|
