; ; File: Log.a ; ; Contains: Routines to emulate logarithmic functions ; ; Originally Written by: Motorola Inc. ; Adapted to Apple/MPW: Jon Okada ; ; Copyright: © 1990,1991 by Apple Computer, Inc., all rights reserved. ; ; This file is used in these builds: Mac32 ; ; Change History (most recent first): ; ; <3> 4/13/91 BG Modified FLOG2 emulation to not signal inexact on exact cases. ; <2> 3/30/91 BG Rolling in Jon Okada's latest changes. ; <1> 12/14/90 BG First checked into TERROR/BBS. ; log.a ; Based upon Motorola files 'slogn.sa' and 'slog2.sa'. ; CHANGE LOG: ; 04 Jan 91 JPO Changed constant names BOUNDS1 and BOUNDS2 to BND1LOG ; and BND2LOG, respectively. Moved all slogn, slog2, and ; slog10 constants and table LOGTBL to file 'constants.a'. ; Changed variable names X, XDCARE, and XFRAC to XLN, ; XLNDC, and XLNFR, respectively. Deleted unreferenced ; label "HiX_0". ; 28 Mar 91 JPO Modified routines "slog2d" and "slog2" to handle exact ; cases of FLOG2. Streamlined "slognd" and "slogn" ; routines (FLOGN). Created a separate subroutine, ; "lognrm", to normalize subnormal input. ; ; slogn * * 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 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 ADJK equ L_SCR1 ;X equ FP_SCR1 ; deleted <1/4/91, JPO> ;XDCARE equ X+2 ;XFRAC equ X+4 XLN equ FP_SCR1 ; <1/4/91, JPO> XLNDC equ XLN+2 XLNFR equ XLN+4 F equ FP_SCR2 FFRAC equ F+4 KLOG2 equ FP_SCR3 SAVEU equ FP_SCR4 slognd: *--ENTRY POINT FOR LOG(X) FOR DENORMALIZED INPUT ; MOVE.L #-100,ADJK(a6) ...INPUT = 2^(ADJK) * FP0 - deleted <3/28/91, JPO> tst.l (a0) ; invalid if negative operand <3/28/91, JPO> bmi t_operr ; <3/28/91, JPO> bsr.b lognrm ; normalize input and initialize ADJK <3/28/91, JPO> bra.b LOGBGN ; continue below <3/28/91, JPO> ; NEW SUBROUTINE - <3/28/91, JPO> ; Subroutine lognrm---normalizes the positive extended denormal input at (a0) and ; writes result with zero exponent to XLN(a6). The negative exponent adjustment ; is written to ADJK(a6). ; On input: a0 points to (assumed positive) input argument. ; On output: XLN(a6) contains normal argument with zero exponent. ; ADJK(a6) contains the negative exponent adjustment. ; a0 points to XLN(a6). *----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. lognrm: ; label added <3/28/91, JPO> thru next ; MoveM.L D2-D7,-(A7) ...save some registers - DELETED <3/28/91, JPO> ; Move.L #$00000000,D3 ...D3 is exponent of smallest norm. # - DELETED <3/28/91, JPO> movem.l D3-D7,-(a7) ; save 5 D registers <3/28/91, JPO> Move.L 4(A0),D4 Move.L 8(A0),D5 ...(D4,D5) is (Hi_X,Lo_X) ; Clr.L D2 ...D2 used for holding K - DELETED <3/28/91, JPO> clr.l d3 ; d3 used for holding k <3/28/91, JPO> ; Tst.L D4 ; DELETED <3/28/91, JPO> ; BNE.B HiX_not0 ; DELETED <3/28/91, JPO> bfffo D4{0:32},D6 ; find first set bit in Hi_X <3/28/91, JPO> bne.b HiX_not0 ; HiX nonzero <3/28/91, JPO> ;HiX_0: ; label DELETED <1/4/91, JPO> Move.L D5,D4 Clr.L D5 ; Move.L #32,D2 ; DELETED <3/28/91, JPO> moveq.l #32,d3 ; <3/28/91, JPO> ; Clr.L D6 ; DELETED <3/28/91, JPO> BFFFO D4{0:32},D6 ; LSL.L D6,D4 ; DELETED <3/28/91, JPO> ; Add.L D6,D2 ...(D3,D4,D5) is normalized - DELETED <3/28/91, JPO> ; Move.L D3,XLN(a6) ; <1/4/91, JPO> - DELETED <3/28/91, JPO> ; Move.L D4,XLNFR(a6) ; <1/4/91, JPO> - DELETED <3/28/91, JPO> ; Move.L D5,XLNFR+4(a6) ; <1/4/91, JPO> - DELETED <3/28/91, JPO> ; Neg.L D2 ; DELETED <3/28/91, JPO> ; Move.L D2,ADJK(a6) ; DELETED <3/28/91, JPO> ; FMove.X XLN(a6),FP0 ; <1/4/91, JPO> - DELETED <3/28/91, JPO> ; MoveM.L (A7)+,D2-D7 ...restore registers - DELETED <3/28/91, JPO> ; LEA XLN(a6),A0 ; <1/4/91, JPO> - DELETED <3/28/91, JPO> ; Bra.B LOGBGN ...begin regular log(X) - DELETED <3/28/91, JPO> HiX_not0: ; Clr.L D6 ; DELETED <3/28/91, JPO> ; BFFFO D4{0:32},D6 ...find first 1 - DELETED <3/28/91, JPO> ; Move.L D6,D2 ...get k - DELETED <3/28/91, JPO> LSL.L D6,D4 add.l d6,d3 ; get k <3/28/91, JPO> Move.L D5,D7 ...a copy of D5 LSL.L D6,D5 ; Neg.L D6 ; DELETED - <3/28/91, JPO> ; AddI.L #32,D6 ; DELETED - <3/28/91, JPO> neg.b d6 ; do byte operations <3/28/91, JPO> addi.b #32,d6 ; <3/28/91, JPO> LSR.L D6,D7 neg.l d3 ; exponent adjust is -k <3/28/91, JPO> Or.L D7,D4 ...(D3,D4,D5) normalized ; Move.L D3,XLN(a6) ; <1/4/91, JPO> - DELETED <3/28/91, JPO> clr.l XLN(a6) ; zero exponent <3/28/91, JPO> Move.L D4,XLNFR(a6) ; <1/4/91, JPO> Move.L D5,XLNFR+4(a6) ; <1/4/91, JPO> ; Neg.L D2 ; DELETED <3/28/91, JPO> ; Move.L D2,ADJK(a6) ; DELETED <3/28/91, JPO> move.l D3,ADJK(a6) ; store exponent adjust <3/28/91, JPO> ; FMove.X XLN(a6),FP0 ; <1/4/91, JPO> - DELETED <3/28/91, JPO> ; MoveM.L (A7)+,D2-D7 ...restore registers - DELETED <3/28/91, JPO> movem.l (a7)+,d3-d7 ; restore 5 registers <3/28/91, JPO> LEA XLN(a6),A0 ; <1/4/91, JPO> ; Bra.B LOGBGN ...begin regular log(X) - DELETED <3/28/91, JPO> rts ; return <3/28/91, JPO> slogn: *--ENTRY POINT FOR LOG(X) FOR X FINITE, NON-ZERO, NOT NAN'S tst.l (a0) ; invalid if negative operand bmi t_operr ; FMOVE.X (A0),FP0 ...LOAD INPUT - moved below <3/28/91, JPO> ; MOVE.L #$00000000,ADJK(a6) ; DELETED <3/28/91, JPO> clr.l ADJK(a6) ; zero exponent adjustment <3/28/91, JPO> move.l (a0),XLN(a6) ; transfer normal operand to XLN(a6) <3/28/91, JPO> move.l 4(a0),XLN+4(a6) ; <3/28/91, JPO> move.l 8(a0),XLN+8(a6) ; <3/28/91, JPO> lea.l XLN(a6),a0 ; a0 points to XLN(a6) <3/28/91, JPO> LOGBGN: *--FPCR SAVED AND CLEARED, INPUT IS 2^(ADJK)*FP0, FP0 CONTAINS *--A FINITE, NON-ZERO, NORMALIZED NUMBER. move.l (a0),d0 move.w 4(a0),d0 FMOVE.X (a0),FP0 ; FPO <- normal operand <3/28/91, JPO> thru next ; move.l (a0),XLN(a6) ; <1/4/91, JPO> - moved to above <3/28/91, JPO> ; move.l 4(a0),XLN+4(a6) ; <1/4/91, JPO> - <3/28/91, JPO> ; move.l 8(a0),XLN+8(a6) ; <1/4/91, JPO> - <3/28/91, JPO> ; CMPI.L #0,D0 ...CHECK IF X IS NEGATIVE - deleted <3/28/91, JPO> ; BLT.W LOGNEG ...LOG OF NEGATIVE ARGUMENT IS INVALID - deleted <3/28/91, JPO> CMP2.L BND1LOG,D0 ...X IS POSITIVE, CHECK IF X IS NEAR 1 BCC.W 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. ASR.L #8,D0 ASR.L #8,D0 ;...SHIFTED 16 BITS, BIASED EXPO. OF X SUBI.L #$3FFF,D0 ;...THIS IS K ADD.L ADJK(a6),D0 ;...ADJUST K, ORIGINAL INPUT MAY BE DENORM. LEA LOGTBL,A0 ;...BASE ADDRESS OF 1/F AND LOG(F) FMOVE.L D0,FP1 ;...CONVERT K TO FLOATING-POINT FORMAT *--WHILE THE CONVERSION IS GOING ON, WE GET F AND ADDRESS OF 1/F MOVE.L #$3FFF0000,XLN(a6) ;...X IS NOW Y, I.E. 2^(-K)*X <1/4/91, JPO> MOVE.L XLNFR(a6),FFRAC(a6) ; <1/4/91, JPO> ANDI.L #$FE000000,FFRAC(a6) ;...FIRST 7 BITS OF Y ORI.L #$01000000,FFRAC(a6) ;...GET F: ATTACH A 1 AT THE EIGHTH BIT MOVE.L FFRAC(a6),D0 ;...READY TO GET ADDRESS OF 1/F ANDI.L #$7E000000,D0 ASR.L #8,D0 ASR.L #8,D0 ASR.L #4,D0 ;...SHIFTED 20, D0 IS THE DISPLACEMENT ADDA.L D0,A0 ;...A0 IS THE ADDRESS FOR 1/F FMOVE.X XLN(a6),FP0 ; <1/4/91, JPO> move.l #$3fff0000,F(a6) clr.l F+8(a6) FSUB.X F(a6),FP0 ;...Y-F FMOVEm.X 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 FMUL.X (A0),FP0 ...FP0 IS U = (Y-F)/F FMUL.X LOGOF2,FP1 ...GET K*LOG2 WHILE FP0 IS NOT READY FMOVE.X FP0,FP2 FMUL.X FP2,FP2 ;...FP2 IS V=U*U FMOVE.X FP1,KLOG2(a6) ;...PUT K*LOG2 IN MEMEORY, 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))] FMOVE.X FP2,FP3 FMOVE.X FP2,FP1 FMUL.D LOGA6,FP1 ;...V*A6 FMUL.D LOGA5,FP2 ;...V*A5 FADD.D LOGA4,FP1 ;...A4+V*A6 FADD.D LOGA3,FP2 ;...A3+V*A5 FMUL.X FP3,FP1 ;...V*(A4+V*A6) FMUL.X FP3,FP2 ;...V*(A3+V*A5) FADD.D LOGA2,FP1 ;...A2+V*(A4+V*A6) FADD.D LOGA1,FP2 ;...A1+V*(A3+V*A5) FMUL.X FP3,FP1 ;...V*(A2+V*(A4+V*A6)) ADDA.L #16,A0 ;...ADDRESS OF LOG(F) FMUL.X FP3,FP2 ;...V*(A1+V*(A3+V*A5)), FP3 RELEASED FMUL.X FP0,FP1 ;...U*V*(A2+V*(A4+V*A6)) FADD.X FP2,FP0 ;...U+V*(A1+V*(A3+V*A5)), FP2 RELEASED FADD.X (A0),FP1 ;...LOG(F)+U*V*(A2+V*(A4+V*A6)) FMOVEm.X (sp)+,FP2/fp3 ;...RESTORE FP2 FADD.X FP1,FP0 ;...FP0 IS LOG(F) + LOG(1+U) fmove.l d1,fpcr FADD.X KLOG2(a6),FP0 ;...FINAL ADD bra t_frcinx LOGNEAR1: *--REGISTERS SAVED: FPCR, FP1. FP0 CONTAINS THE INPUT. FMOVE.X FP0,FP1 FSUB.S one,FP1 ;...FP1 IS X-1 FADD.S one,FP0 ;...FP0 IS X+1 FADD.X 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 FDIV.X FP0,FP1 ;...FP1 IS U FMOVEm.X 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)] ) FMOVE.X FP1,FP0 FMUL.X FP0,FP0 ;...FP0 IS V FMOVE.X FP1,SAVEU(a6) ;...STORE U IN MEMORY, FREE FP1 FMOVE.X FP0,FP1 FMUL.X FP1,FP1 ;...FP1 IS W FMOVE.D LOGB5,FP3 FMOVE.D LOGB4,FP2 FMUL.X FP1,FP3 ;...W*B5 FMUL.X FP1,FP2 ;...W*B4 FADD.D LOGB3,FP3 ;...B3+W*B5 FADD.D LOGB2,FP2 ;...B2+W*B4 FMUL.X FP3,FP1 ;...W*(B3+W*B5), FP3 RELEASED FMUL.X FP0,FP2 ;...V*(B2+W*B4) FADD.D LOGB1,FP1 ;...B1+W*(B3+W*B5) FMUL.X SAVEU(a6),FP0 ;...FP0 IS U*V FADD.X FP2,FP1 ;...B1+W*(B3+W*B5) + V*(B2+W*B4), FP2 RELEASED FMOVEm.X (sp)+,FP2/fp3 ;...FP2 RESTORED FMUL.X FP1,FP0 ;...U*V*( [B1+W*(B3+W*B5)] + [V*(B2+W*B4)] ) fmove.l d1,fpcr FADD.X SAVEU(a6),FP0 bra t_frcinx rts ;LOGNEG: ; label DELETED <3/28/91, JPO> ;*--REGISTERS SAVED FPCR. LOG(-VE) IS INVALID ; bra t_operr ; DELETED <3/28/91, JPO> slognp1d: *--ENTRY POINT FOR LOG(1+Z) FOR DENORMALIZED INPUT * Simply return the denorm bra t_extdnrm slognp1: *--ENTRY POINT FOR LOG(1+X) FOR X FINITE, NON-ZERO, NOT NAN'S FMOVE.X (A0),FP0 ...LOAD INPUT fabs.x fp0 ;test magnitude fcmp.x LTHOLD,fp0 ;compare with min threshold fbgt.w LP1REAL ;if greater, continue fmove.l #0,fpsr ;clr N flag from compare fmove.l d1,fpcr fmove.x (a0),fp0 ;return signed argument bra t_frcinx LP1REAL: FMOVE.X (A0),FP0 ...LOAD INPUT MOVE.L #$00000000,ADJK(a6) FMOVE.X FP0,FP1 ...FP1 IS INPUT Z FADD.S one,FP0 ...X := ROUND(1+Z) FMOVE.X FP0,XLN(a6) ; <1/4/91, JPO> MOVE.W XLNFR(a6),XLNDC(a6) ; <1/4/91, JPO> MOVE.L XLN(a6),D0 ; <1/4/91, JPO> ; CMPI.L #0,D0 ; DELETED <3/28/91, JPO> tst.l d0 ; <3/28/91, JPO> BLE.W LP1NEG0 ...LOG OF ZERO OR -VE CMP2.L BND2LOG,D0 ; BCS.W LOGMAIN ...BND2LOG IS [1/2,3/2] *--IF 1+Z > 3/2 OR 1+Z < 1/2, THEN X, WHICH IS ROUNDING 1+Z, *--CONTAINS AT LEAST 63 BITS OF INFORMATION OF Z. IN THAT CASE, *--SIMPLY INVOKE LOG(X) FOR LOG(1+Z). LP1NEAR1: *--NEXT SEE IF EXP(-1/16) < X < EXP(1/16) CMP2.L BND1LOG,D0 BCS.B LP1CARE LP1ONE16: *--EXP(-1/16) < X < EXP(1/16). LOG(1+Z) = LOG(1+U/2) - LOG(1-U/2) *--WHERE U = 2Z/(2+Z) = 2Z/(1+X). FADD.X FP1,FP1 ...FP1 IS 2Z FADD.S one,FP0 ...FP0 IS 1+X *--U = FP1/FP0 BRA.W LP1CONT2 LP1CARE: *--HERE WE USE THE USUAL TABLE DRIVEN APPROACH. CARE HAS TO BE *--TAKEN BECAUSE 1+Z CAN HAVE 67 BITS OF INFORMATION AND WE MUST *--PRESERVE ALL THE INFORMATION. BECAUSE 1+Z IS IN [1/2,3/2], *--THERE ARE ONLY TWO CASES. *--CASE 1: 1+Z < 1, THEN K = -1 AND Y-F = (2-F) + 2Z *--CASE 2: 1+Z > 1, THEN K = 0 AND Y-F = (1-F) + Z *--ON RETURNING TO LP1CONT1, WE MUST HAVE K IN FP1, ADDRESS OF *--(1/F) IN A0, Y-F IN FP0, AND FP2 SAVED. MOVE.L XLNFR(a6),FFRAC(a6) ; <1/4/91, JPO> ANDI.L #$FE000000,FFRAC(a6) ORI.L #$01000000,FFRAC(a6) ;...F OBTAINED CMPI.L #$3FFF8000,D0 ;...SEE IF 1+Z > 1 BGE.B KISZERO KISNEG1: FMOVE.S TWO,FP0 move.l #$3FFF0000,F(a6) clr.l F+8(a6) FSUB.X F(a6),FP0 ;...2-F MOVE.L FFRAC(a6),D0 ANDI.L #$7E000000,D0 ASR.L #8,D0 ASR.L #8,D0 ASR.L #4,D0 ;...D0 CONTAINS DISPLACEMENT FOR 1/F FADD.X FP1,FP1 ;...GET 2Z FMOVEm.X FP2/fp3,-(sp) ;...SAVE FP2 FADD.X FP1,FP0 ;...FP0 IS Y-F = (2-F)+2Z LEA LOGTBL,A0 ;...A0 IS ADDRESS OF 1/F ADDA.L D0,A0 FMOVE.S negone,FP1 ;...FP1 IS K = -1 BRA.W LP1CONT1 KISZERO: FMOVE.S one,FP0 move.l #$3fff0000,F(a6) clr.l F+8(a6) FSUB.X F(a6),FP0 ;...1-F MOVE.L FFRAC(a6),D0 ANDI.L #$7E000000,D0 ASR.L #8,D0 ASR.L #8,D0 ASR.L #4,D0 FADD.X FP1,FP0 ;...FP0 IS Y-F FMOVEm.X FP2/fp3,-(sp) ;...FP2 SAVED LEA LOGTBL,A0 ADDA.L D0,A0 ;...A0 IS ADDRESS OF 1/F FMOVE.S zero,FP1 ;...FP1 IS K = 0 BRA.W LP1CONT1 LP1NEG0: *--FPCR SAVED. D0 IS X IN COMPACT FORM. ; CMPI.L #0,D0 ; DELETED <3/28/91, JPO> ; BLT.B LP1NEG ; DELETED <3/28/91, JPO> tst.l d0 ; <3/28/91, JPO> bmi.b LP1NEG ; <3/28/91, JPO> LP1ZERO: FMOVE.S negone,FP0 fmove.l d1,fpcr bra t_dz LP1NEG: FMOVE.S zero,FP0 fmove.l d1,fpcr bra t_operr ; slog2 * * slog2.sa 3.1 12/10/90 * * The entry point slog10 computes the base-10 * logarithm of an input argument X. * slog10d does the same except the input value is a * denormalized number. * sLog2 and sLog2d are the base-2 analogues. * * INPUT: Double-extended value in memory location pointed to * by address register a0. * * OUTPUT: log_10(X) or log_2(X) returned in floating-point * register fp0. * * ACCURACY and MONOTONICITY: The returned result is within 1.7 * ulps in 64 significant bit, i.e. within 0.5003 ulp * to 53 bits if the result is subsequently rounded * to double precision. The result is provably monotonic * in double precision. * * SPEED: Two timings are measured, both in the copy-back mode. * The first one is measured when the function is invoked * the first time (so the instructions and data are not * in cache), and the second one is measured when the * function is reinvoked at the same input argument. * * ALGORITHM and IMPLEMENTATION NOTES: * * slog10d: * * Step 0. If X < 0, create a NaN and raise the invalid operation * flag. Otherwise, save FPCR in D1; set FpCR to default. * Notes: Default means round-to-nearest mode, no floating-point * traps, and precision control = double extended. * * Step 1. Call slognd to obtain Y = log(X), the natural log of X. * Notes: Even if X is denormalized, log(X) is always normalized. * * Step 2. Compute log_10(X) = log(X) * (1/log(10)). * 2.1 Restore the user FPCR * 2.2 Return ans := Y * INV_L10. * * * slog10: * * Step 0. If X < 0, create a NaN and raise the invalid operation * flag. Otherwise, save FPCR in D1; set FpCR to default. * Notes: Default means round-to-nearest mode, no floating-point * traps, and precision control = double extended. * * Step 1. Call sLogN to obtain Y = log(X), the natural log of X. * * Step 2. Compute log_10(X) = log(X) * (1/log(10)). * 2.1 Restore the user FPCR * 2.2 Return ans := Y * INV_L10. * * * sLog2d: * * Step 0. If X < 0, create a NaN and raise the invalid operation * flag. Otherwise, save FPCR in D1; set FpCR to default. * Notes: Default means round-to-nearest mode, no floating-point * traps, and precision control = double extended. * * Step 1. Call slognd to obtain Y = log(X), the natural log of X. * Notes: Even if X is denormalized, log(X) is always normalized. * * Step 2. Compute log_2(X) = log(X) * (1/log(2)). * 2.1 Restore the user FPCR * 2.2 Return ans := Y * INV_L2. * * * sLog2: * * Step 0. If X < 0, create a NaN and raise the invalid operation * flag. Otherwise, save FPCR in D1; set FpCR to default. * Notes: Default means round-to-nearest mode, no floating-point * traps, and precision control = double extended. * * Step 1. If X is not an integer power of two, i.e., X != 2^k, * go to Step 3. * * Step 2. Return k. * 2.1 Get integer k, X = 2^k. * 2.2 Restore the user FPCR. * 2.3 Return ans := convert-to-double-extended(k). * * Step 3. Call sLogN to obtain Y = log(X), the natural log of X. * * Step 4. Compute log_2(X) = log(X) * (1/log(2)). * 4.1 Restore the user FPCR * 4.2 Return ans := Y * INV_L2. * * 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. * SLOG2 IDNT 2,1 Motorola 040 Floating Point Software Package slog10d: *--entry point for Log10(X), X is denormalized ; move.l (a0),d0 ; DELETED <3/28/91, JPO> thru next ; blt.w invalid ; DELETED <3/28/91, JPO> tst.l (a0) ; <3/28/91, JPO> bmi t_operr ; <3/28/91, JPO> move.l d1,-(sp) clr.l d1 bsr slognd ...log(X), X denorm. fmove.l (sp)+,fpcr fmul.x INV_L10,fp0 bra t_frcinx slog10: *--entry point for Log10(X), X is normalized ; move.l (a0),d0 ; DELETED <3/28/91, JPO> thru next ; blt.b invalid ; DELETED <3/28/91, JPO> tst.l (a0) ; <3/28/91, JPO> bmi t_operr ; <3/28/91, JPO> move.l d1,-(sp) clr.l d1 bsr slogn ...log(X), X normal. fmove.l (sp)+,fpcr fmul.x INV_L10,fp0 bra t_frcinx slog2d: *--entry point for Log2(X), X is denormalized ; move.l (a0),d0 ; DELETED <3/28/91, JPO> thru next ; blt.b invalid ; DELETED <3/28/91, JPO> tst.l (a0) ; <3/28/91, JPO> bmi t_operr ; <3/28/91, JPO> bsr lognrm ; normalize with exponent adjust in ADJK(A6) <3/28/91, JPO> tst.l 8(a0) ; check for exact integral power of 2 <3/28/91, JPO> bne.b @1 ; inexact <3/28/91, JPO> move.l 4(a0),d0 ; <3/28/91, JPO> lsl.l #1,d0 ; <3/28/91, JPO> bne.b @1 ; inexact <3/28/91, JPO> move.l ADJK(a6),d0 ; exact. get negative exponent into d0 <3/28/91, JPO> bra.b slog2ex ; continue below <3/28/91, JPO> @1: ; label added <3/28/91, JPO> move.l d1,-(sp) clr.l d1 ; bsr slognd ...log(X), X denorm. - DELETED <3/28/91, JPO> bsr LOGBGN ; log(X), X norm with negative ADJK fmove.l (sp)+,fpcr fmul.x INV_L2,fp0 bra t_frcinx slog2: *--entry point for Log2(X), X is normalized ; move.l (a0),d0 ; DELETED <3/28/91, JPO> thru next ; blt.b invalid ; DELETED <3/28/91, JPO> tst.l (a0) ; <3/28/91, JPO> bmi t_operr ; <3/28/91, JPO> ; move.l 8(a0),d0 ; DELETED <3/28/91, JPO> tst.l 8(a0) bne.b continue ...X is not 2^k move.l 4(a0),d0 and.l #$7FFFFFFF,d0 ; tst.l d0 ; DELETED <3/28/91, JPO> bne.b continue *--X = 2^k. move.w (a0),d0 slog2ex: ; label ADDED <3/28/91, JPO> thru next ; and.l #$00007FFF,d0 ; DELETED <3/28/91, JPO> ; sub.l #$3FFF,d0 ; DELETED <3/28/91, JPO> sub.w #$3FFF,d0 ; word operation sufficient <3/28/91, JPO> fmove.l d1,fpcr ; fmove.l d0,fp0 ; DELETED <3/28/91, JPO> fmove.w d0,fp0 ; exact result <3/28/91, JPO> ; bra t_frcinx ; DELETED rts ; return exact result <3/28/91, JPO> continue: move.l d1,-(sp) clr.l d1 bsr slogn ;...log(X), X normal. fmove.l (sp)+,fpcr fmul.x INV_L2,fp0 bra t_frcinx ;invalid: ; label DELETED <3/28/91, JPO> ; bra t_operr ; DELETED <3/28/91, JPO>