|
|
| stwotox.sa 3.1 12/10/90
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|
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| stwotox --- 2**X
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| stwotoxd --- 2**X for denormalized X
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| stentox --- 10**X
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| stentoxd --- 10**X for denormalized X
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|
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| Input: Double-extended number X in location pointed to
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| by address register a0.
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|
|
| Output: The function values are returned in Fp0.
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|
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| 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
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| result is subsequently rounded to double precision. The
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| result is provably monotonic in double precision.
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|
|
| Speed: The program stwotox takes approximately 190 cycles and the
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| program stentox takes approximately 200 cycles.
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|
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| Algorithm:
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|
|
| twotox
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| 1. If |X| > 16480, go to ExpBig.
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|
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| 2. If |X| < 2**(-70), go to ExpSm.
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|
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| 3. Decompose X as X = N/64 + r where |r| <= 1/128. Furthermore
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| decompose N as
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| N = 64(M + M') + j, j = 0,1,2,...,63.
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|
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| 4. Overwrite r := r * log2. Then
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| 2**X = 2**(M') * 2**(M) * 2**(j/64) * exp(r).
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| Go to expr to compute that expression.
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|
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| tentox
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| 1. If |X| > 16480*log_10(2) (base 10 log of 2), go to ExpBig.
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|
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| 2. If |X| < 2**(-70), go to ExpSm.
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|
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| 3. Set y := X*log_2(10)*64 (base 2 log of 10). Set
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| N := round-to-int(y). Decompose N as
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| N = 64(M + M') + j, j = 0,1,2,...,63.
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|
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| 4. Define r as
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| r := ((X - N*L1)-N*L2) * L10
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| where L1, L2 are the leading and trailing parts of log_10(2)/64
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| and L10 is the natural log of 10. Then
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| 10**X = 2**(M') * 2**(M) * 2**(j/64) * exp(r).
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| Go to expr to compute that expression.
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|
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| expr
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| 1. Fetch 2**(j/64) from table as Fact1 and Fact2.
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|
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| 2. Overwrite Fact1 and Fact2 by
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| Fact1 := 2**(M) * Fact1
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| Fact2 := 2**(M) * Fact2
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| Thus Fact1 + Fact2 = 2**(M) * 2**(j/64).
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|
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| 3. Calculate P where 1 + P approximates exp(r):
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| P = r + r*r*(A1+r*(A2+...+r*A5)).
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|
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| 4. Let AdjFact := 2**(M'). Return
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| AdjFact * ( Fact1 + ((Fact1*P) + Fact2) ).
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| Exit.
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|
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| ExpBig
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| 1. Generate overflow by Huge * Huge if X > 0; otherwise, generate
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| underflow by Tiny * Tiny.
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|
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| ExpSm
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| 1. Return 1 + X.
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|
|
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| Copyright (C) Motorola, Inc. 1990
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| All Rights Reserved
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|
|
| For details on the license for this file, please see the
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| file, README, in this same directory.
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|
|STWOTOX idnt 2,1 | Motorola 040 Floating Point Software Package
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|
|section 8
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|
#include "fpsp.h"
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|
BOUNDS1: .long 0x3FB98000,0x400D80C0 | ... 2^(-70),16480
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BOUNDS2: .long 0x3FB98000,0x400B9B07 | ... 2^(-70),16480 LOG2/LOG10
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|
L2TEN64: .long 0x406A934F,0x0979A371 | ... 64LOG10/LOG2
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L10TWO1: .long 0x3F734413,0x509F8000 | ... LOG2/64LOG10
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|
L10TWO2: .long 0xBFCD0000,0xC0219DC1,0xDA994FD2,0x00000000
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|
LOG10: .long 0x40000000,0x935D8DDD,0xAAA8AC17,0x00000000
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|
LOG2: .long 0x3FFE0000,0xB17217F7,0xD1CF79AC,0x00000000
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|
EXPA5: .long 0x3F56C16D,0x6F7BD0B2
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EXPA4: .long 0x3F811112,0x302C712C
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EXPA3: .long 0x3FA55555,0x55554CC1
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EXPA2: .long 0x3FC55555,0x55554A54
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EXPA1: .long 0x3FE00000,0x00000000,0x00000000,0x00000000
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|
HUGE: .long 0x7FFE0000,0xFFFFFFFF,0xFFFFFFFF,0x00000000
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TINY: .long 0x00010000,0xFFFFFFFF,0xFFFFFFFF,0x00000000
|
|
EXPTBL:
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.long 0x3FFF0000,0x80000000,0x00000000,0x3F738000
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.long 0x3FFF0000,0x8164D1F3,0xBC030773,0x3FBEF7CA
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.long 0x3FFF0000,0x82CD8698,0xAC2BA1D7,0x3FBDF8A9
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.long 0x3FFF0000,0x843A28C3,0xACDE4046,0x3FBCD7C9
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.long 0x3FFF0000,0x85AAC367,0xCC487B15,0xBFBDE8DA
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.long 0x3FFF0000,0x871F6196,0x9E8D1010,0x3FBDE85C
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.long 0x3FFF0000,0x88980E80,0x92DA8527,0x3FBEBBF1
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.long 0x3FFF0000,0x8A14D575,0x496EFD9A,0x3FBB80CA
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.long 0x3FFF0000,0x8B95C1E3,0xEA8BD6E7,0xBFBA8373
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.long 0x3FFF0000,0x8D1ADF5B,0x7E5BA9E6,0xBFBE9670
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.long 0x3FFF0000,0x8EA4398B,0x45CD53C0,0x3FBDB700
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.long 0x3FFF0000,0x9031DC43,0x1466B1DC,0x3FBEEEB0
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.long 0x3FFF0000,0x91C3D373,0xAB11C336,0x3FBBFD6D
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.long 0x3FFF0000,0x935A2B2F,0x13E6E92C,0xBFBDB319
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.long 0x3FFF0000,0x94F4EFA8,0xFEF70961,0x3FBDBA2B
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.long 0x3FFF0000,0x96942D37,0x20185A00,0x3FBE91D5
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.long 0x3FFF0000,0x9837F051,0x8DB8A96F,0x3FBE8D5A
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.long 0x3FFF0000,0x99E04593,0x20B7FA65,0xBFBCDE7B
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.long 0x3FFF0000,0x9B8D39B9,0xD54E5539,0xBFBEBAAF
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.long 0x3FFF0000,0x9D3ED9A7,0x2CFFB751,0xBFBD86DA
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.long 0x3FFF0000,0x9EF53260,0x91A111AE,0xBFBEBEDD
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.long 0x3FFF0000,0xA0B0510F,0xB9714FC2,0x3FBCC96E
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.long 0x3FFF0000,0xA2704303,0x0C496819,0xBFBEC90B
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.long 0x3FFF0000,0xA43515AE,0x09E6809E,0x3FBBD1DB
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.long 0x3FFF0000,0xA5FED6A9,0xB15138EA,0x3FBCE5EB
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.long 0x3FFF0000,0xA7CD93B4,0xE965356A,0xBFBEC274
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.long 0x3FFF0000,0xA9A15AB4,0xEA7C0EF8,0x3FBEA83C
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.long 0x3FFF0000,0xAB7A39B5,0xA93ED337,0x3FBECB00
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.long 0x3FFF0000,0xAD583EEA,0x42A14AC6,0x3FBE9301
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.long 0x3FFF0000,0xAF3B78AD,0x690A4375,0xBFBD8367
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.long 0x3FFF0000,0xB123F581,0xD2AC2590,0xBFBEF05F
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.long 0x3FFF0000,0xB311C412,0xA9112489,0x3FBDFB3C
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.long 0x3FFF0000,0xB504F333,0xF9DE6484,0x3FBEB2FB
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.long 0x3FFF0000,0xB6FD91E3,0x28D17791,0x3FBAE2CB
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.long 0x3FFF0000,0xB8FBAF47,0x62FB9EE9,0x3FBCDC3C
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.long 0x3FFF0000,0xBAFF5AB2,0x133E45FB,0x3FBEE9AA
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.long 0x3FFF0000,0xBD08A39F,0x580C36BF,0xBFBEAEFD
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.long 0x3FFF0000,0xBF1799B6,0x7A731083,0xBFBCBF51
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.long 0x3FFF0000,0xC12C4CCA,0x66709456,0x3FBEF88A
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.long 0x3FFF0000,0xC346CCDA,0x24976407,0x3FBD83B2
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.long 0x3FFF0000,0xC5672A11,0x5506DADD,0x3FBDF8AB
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.long 0x3FFF0000,0xC78D74C8,0xABB9B15D,0xBFBDFB17
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.long 0x3FFF0000,0xC9B9BD86,0x6E2F27A3,0xBFBEFE3C
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.long 0x3FFF0000,0xCBEC14FE,0xF2727C5D,0xBFBBB6F8
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.long 0x3FFF0000,0xCE248C15,0x1F8480E4,0xBFBCEE53
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.long 0x3FFF0000,0xD06333DA,0xEF2B2595,0xBFBDA4AE
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.long 0x3FFF0000,0xD2A81D91,0xF12AE45A,0x3FBC9124
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.long 0x3FFF0000,0xD4F35AAB,0xCFEDFA1F,0x3FBEB243
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.long 0x3FFF0000,0xD744FCCA,0xD69D6AF4,0x3FBDE69A
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.long 0x3FFF0000,0xD99D15C2,0x78AFD7B6,0xBFB8BC61
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.long 0x3FFF0000,0xDBFBB797,0xDAF23755,0x3FBDF610
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.long 0x3FFF0000,0xDE60F482,0x5E0E9124,0xBFBD8BE1
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.long 0x3FFF0000,0xE0CCDEEC,0x2A94E111,0x3FBACB12
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.long 0x3FFF0000,0xE33F8972,0xBE8A5A51,0x3FBB9BFE
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.long 0x3FFF0000,0xE5B906E7,0x7C8348A8,0x3FBCF2F4
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.long 0x3FFF0000,0xE8396A50,0x3C4BDC68,0x3FBEF22F
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.long 0x3FFF0000,0xEAC0C6E7,0xDD24392F,0xBFBDBF4A
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.long 0x3FFF0000,0xED4F301E,0xD9942B84,0x3FBEC01A
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.long 0x3FFF0000,0xEFE4B99B,0xDCDAF5CB,0x3FBE8CAC
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.long 0x3FFF0000,0xF281773C,0x59FFB13A,0xBFBCBB3F
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.long 0x3FFF0000,0xF5257D15,0x2486CC2C,0x3FBEF73A
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.long 0x3FFF0000,0xF7D0DF73,0x0AD13BB9,0xBFB8B795
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.long 0x3FFF0000,0xFA83B2DB,0x722A033A,0x3FBEF84B
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.long 0x3FFF0000,0xFD3E0C0C,0xF486C175,0xBFBEF581
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|
.set N,L_SCR1
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|
.set X,FP_SCR1
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.set XDCARE,X+2
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.set XFRAC,X+4
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|
.set ADJFACT,FP_SCR2
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|
.set FACT1,FP_SCR3
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.set FACT1HI,FACT1+4
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.set FACT1LOW,FACT1+8
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|
.set FACT2,FP_SCR4
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.set FACT2HI,FACT2+4
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.set FACT2LOW,FACT2+8
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|
| xref t_unfl
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|xref t_ovfl
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|xref t_frcinx
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|
.global stwotoxd
|
stwotoxd:
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|--ENTRY POINT FOR 2**(X) FOR DENORMALIZED ARGUMENT
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|
fmovel %d1,%fpcr | ...set user's rounding mode/precision
|
fmoves #0x3F800000,%fp0 | ...RETURN 1 + X
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movel (%a0),%d0
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orl #0x00800001,%d0
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fadds %d0,%fp0
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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
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|
movel (%a0),%d0
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movew 4(%a0),%d0
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fmovex %fp0,X(%a6)
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andil #0x7FFFFFFF,%d0
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|
cmpil #0x3FB98000,%d0 | ...|X| >= 2**(-70)?
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bges TWOOK1
|
bra EXPBORS
|
|
TWOOK1:
|
cmpil #0x400D80C0,%d0 | ...|X| > 16480?
|
bles TWOMAIN
|
bra EXPBORS
|
|
|
TWOMAIN:
|
|--USUAL CASE, 2^(-70) <= |X| <= 16480
|
|
fmovex %fp0,%fp1
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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)
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movel (%a1)+,FACT1HI(%a6)
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movel (%a1)+,FACT1LOW(%a6)
|
movew (%a1)+,FACT2(%a6)
|
clrw FACT2+2(%a6)
|
|
fsubx %fp1,%fp0 | ...X - (1/64)*INT(64 X)
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|
movew (%a1)+,FACT2HI(%a6)
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clrw FACT2HI+2(%a6)
|
clrl FACT2LOW(%a6)
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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
|
