| | |
| | 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| | THIS IS UNPUBLISHED PROPRIETARY SOURCE CODE OF MOTOROLA |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 |
