mirror of
https://github.com/elliotnunn/sys7.1-doc-wip.git
synced 2025-02-08 20:31:05 +00:00
228 lines
10 KiB
Plaintext
228 lines
10 KiB
Plaintext
;
|
|
; File: HWElemsTg.a
|
|
;
|
|
; Contains: Routines to calculate TAN(x)
|
|
;
|
|
; Written by: Apple Numerics Group, DSG
|
|
;
|
|
; Copyright: © 1985-1993 by Apple Computer, Inc., all rights reserved.
|
|
;
|
|
; Change History (most recent first):
|
|
;
|
|
; <SM2> 2/3/93 CSS Update from Horror:
|
|
; <H2> 9/29/92 BG Adding Jon Okada's latest fixes.
|
|
; <1> 10/24/91 SAM/KSM Rolled in Regatta file.
|
|
;
|
|
; Terror Change History:
|
|
;
|
|
; <1> 01/06/91 BG Added to TERROR/BBS for the time.
|
|
;
|
|
|
|
;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;
|
|
;; ;;
|
|
;; File 881ELEMStg.a, ;;
|
|
;; Function tg(x), ;;
|
|
;; Implementation of tg for ΩSANE. ;;
|
|
;; For machines with MC68881/68882 FPU. ;;
|
|
;; ;;
|
|
;; Copyright Apple Computer, Inc. 1985-7, 1989, 1990, 1992, 1993. ;;
|
|
;; ;;
|
|
;; Written by Ali Sazegari, started on July 13 1989, ;;
|
|
;; ;;
|
|
;; Started testing on december 11, 1989. All flags are set correctly, ;;
|
|
;; except for small normalized numbers. One intersting fact: in the ;;
|
|
;; denormal case the underflow flag is automatically raised. The mystery ;;
|
|
;; about the 1/tg(x) when tg(x) = 0 still remains! It does not seem to ever ;;
|
|
;; happen. ;;
|
|
;; ;;
|
|
;; December 12, 1989. Fixed all the flags for denormals, small numbers and ;;
|
|
;; edge cases. tg now passes a bag full of test vectors with the correct ;;
|
|
;; flags. ;;
|
|
;; ;;
|
|
;; January 29, 1990. C. McMaster found the 1-ulp error which was occuring ;;
|
|
;; on a small portion of the test cases. The order of operation of the ;;
|
|
;; statment: t^5 * P(t*t)/Q(t*t) was not right. ;;
|
|
;; ;;
|
|
;; September 17, 1990. The mystery about 1/tg(x) is solved ( refer to the ;;
|
|
;; section below ). ±π/2 arguments now give ±∞. All the rest of the hex ;;
|
|
;; inputs for tg in the test file which were commented out work fine. ;;
|
|
;; ;;
|
|
;; Based on Elems881, package code for Macintosh by J. Coonen, ;;
|
|
;; who wrote it from pascal programs by D. Hough, C. McMaster and K. Hanson. ;;
|
|
;; ;;
|
|
;; Further Modification History: ;;
|
|
;; 23 Apr 92 Performance tuning done (better register usage, faster FPU ;;
|
|
;; ops used, front-end check for small operands eliminates ;;
|
|
;; spurious underflowing. ... JPO ;;
|
|
;; ;;
|
|
;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;
|
|
|
|
Rtan
|
|
|
|
;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;
|
|
;; Classification of the argument with a neat trick. ;;
|
|
;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;
|
|
|
|
FMOVE.L FPSR,D0 ; what kind of arg? fpsr has the key
|
|
|
|
;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;
|
|
;; Here is the strategy ( commonly known as the scoop ) by C. R. Lewis: ;;
|
|
;; Register D0 has N, Z, I, NaN burried in it. Mask off unwanted bits to ;;
|
|
;; get them. Then, left shift NZINaN to have Z at MSB position. If D0 = 0 ;;
|
|
;; then arg is finite. ;;
|
|
;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;
|
|
|
|
; MOVE.L #ExcMask,D1 ; mask it kid DELETED <4/23/92, JPO>
|
|
; AND.L D1,D0 ; we now have the meat DELETED <4/23/92, JPO>
|
|
ANDI.L #ExcMask,D0
|
|
LSL.L #ShiftIt,D0 ; N->C, Z->MSB, I->30th & NaN->29th bit
|
|
|
|
;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;
|
|
;; If arg is a non zero finite number, great! let's fall through, otherwise ;;
|
|
;; take the problem cases. ;;
|
|
;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;
|
|
|
|
BNE.W BadBoys ; take care of the bad guys (Z bar set)
|
|
|
|
; MOVEM.L A1,-(SP) ; use a1 for denormal flag use DELETED <4/23/92, JPO>
|
|
; MOVEA.L A0,A1 ; keep a copy of the pointer to argument DELETED <4/23/92, JPO>
|
|
|
|
BFEXTU (A0){1:15},D0 ; |arg| < 2^(-32)? <4/23/92, JPO>
|
|
CMPI.W #$3FDF,D0 ; <4/23/92, JPO>
|
|
BGE.B @tanred ; too big, reduce argument <4/23/92, JPO>
|
|
|
|
MOVE.L #SetInexact,D0 ; small, signal INEXACT <4/23/92, JPO>
|
|
BFTST (A0){1:16} ; denorm signals UNDERFLOW <4/23/92, JPO>
|
|
BNE.B @tinydone ; <4/23/92, JPO>
|
|
OR.L #SetUFlow,D0 ; <4/23/92, JPO>
|
|
@tinydone: ; label ADDED <4/23/92, JPO>
|
|
JMP (A4) ; return arg as result <4/23/92, JPO>
|
|
|
|
|
|
;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;
|
|
;; About the argument reduction: t is reduced REM approximate Pi/2, leaving ;;
|
|
;; its magnitude no bigger than approximate Pi/4. ;;
|
|
;; ;;
|
|
;; Recall that: ;;
|
|
;; ;;
|
|
;; tg(pi/2 + t) = -1/tg(t) ;;
|
|
;; tg(pi + t) = tg(t) ;;
|
|
;; ;;
|
|
;; Then if input t = q*(Pi/2) + r, ;;
|
|
;; q MOD 2 determines whether to negate and reciprocate tg(t). ;;
|
|
;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;
|
|
|
|
@tanred: ; label ADDED <4/23/92, JPO>
|
|
; FMOVE.X FPKPI2,FP1 ; get the approximate Pi/2 DELETED <4/23/92, JPO>
|
|
; FREM.X FP1,FP0 ; t = arg REM Pi/2, quotient q in fpsr DELETED <4/23/92, JPO>
|
|
FREM.X FPKPI2,FP0 ; t = arg REM π/2, quo q in FPSR <4/23/92, JPO>
|
|
|
|
;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;
|
|
;; Evaluate tg(t) for |t| <= pi/4. ;;
|
|
;; ;;
|
|
;; Use the change of variable s = t*t ;;
|
|
;; ;;
|
|
;; if ( s <= 1/4 ) then tg(t) = t + ( t^3/3 + t^5 * P(s)/Q(s) ) ;;
|
|
;; else t' = (t - 3/4)/3 ;;
|
|
;; tg(t) = t + s/4 + s * t' + s * ( t^3 * P(s)/Q(s) ) ;;
|
|
;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;
|
|
|
|
FMOVE.X FP0,FP3 ; fp0 = t, fp1 = Pi/2, fp3 = t
|
|
FMUL.X FP0,FP0 ; fp0 = t*t, fp1 = Pi/2, fp3 = t
|
|
|
|
LEA TANP,A0 ; A0 = coeff table ptr for PolyEval
|
|
BSR PolyEval ; fp1 = PolyEval(P, t*t)
|
|
|
|
FMOVE.X FP1,FP2 ; save P(arg)
|
|
|
|
LEA TANQ,A0 ; A0 = coeff table ptr for PolyEval
|
|
BSR PolyEval ; fp1 = PolyEval(Q, t*t)
|
|
|
|
FDIV.X FP1,FP2 ; form fp2 = P(t*t)/Q(t*t)
|
|
FMUL.X FP0,FP2 ; form fp2 = (t*t) * P(t*t)/Q(t*t)
|
|
FMOVE.X FP0,FP1 ; fp1 = fp0 = t*t
|
|
FMUL.X FP3,FP1 ; form fp1 = t^3
|
|
FMUL.X FP1,FP2 ; form fp2 = t^5 * P(t*t)/Q(t*t)
|
|
|
|
FCMP.S #"0.25",FP0 ; decide which formula to continue
|
|
FBGT.W BigBabe ; well, we decided to use the big babe
|
|
|
|
FMUL.X FP3,FP0 ; compute fp0 = t^3
|
|
; FDIV.W #3,FP0 ; we have fp0 = t^3/3 DELETED <4/23/92, JPO>
|
|
FDIV.S #"$40400000",FP0 ; FP0 <- t^3/3.0 <4/23/92, JPO>
|
|
FADD.X FP2,FP0 ; almost got it fp0 = t^3/3 + t^5 * P/Q
|
|
FADD.X FP3,FP0 ; fp0 = t + t^3/3 + t^5 * P(t*t)/Q(t*t)
|
|
BRA.B LastCheck ; see if it needs more massaging
|
|
|
|
BigBabe
|
|
|
|
FMOVE.X FP3,FP1 ; fp0 = t*t fp1=t fp2 = t^5*P/Q fp3 = t
|
|
; FADD.X #"-0.75",FP1 ; fp1 = t - 3/4 DELETED <4/23/92, JPO>
|
|
; FDIV.W #3,FP1 ; fp1 = t' = ( t - 3/4 ) / 3 DELETED <4/23/92, JPO>
|
|
FADD.S #"$BF400000",FP1 ; FP1 <- t - 0.75 <4/23/92, JPO>
|
|
FDIV.S #"$40400000",FP1 ; FP1 <- t' = (t - 0.75)/3.0 <4/23/92, JPO>
|
|
FMUL.X FP0,FP1 ; fp1 = t' * s = t' * t * t
|
|
FADD.X FP2,FP1 ; fp1 = t' * s + t^5 * P(s)/Q(s)
|
|
; FDIV.W #4,FP0 ; before fp0 = t*t, now t*t/4 DELETED <4/23/92, JPO>
|
|
FMUL.S #"$3E800000",FP0 ; FP0 <- t*t*0.25 <4/23/92, JPO>
|
|
FADD.X FP1,FP0 ; fp0 = s/4 + s*t' + s*(t^3*P(s)/Q(s))
|
|
FADD.X FP3,FP0 ; finished, this is tg(t)
|
|
|
|
LastCheck
|
|
|
|
FMOVE.L FPSR,D0 ; q is in 1st byte of higher word
|
|
SWAP D0 ; q is in 1st byte of lower word
|
|
LSR.B #1,D0 ; is it 0 or 1? ( odd or even )
|
|
BCC.B @1 ; ok, get ready to fly out
|
|
; FMOVE.W #-1,FP1 ; it is odd so invert & negate tg(t) DELETED <4/23/92, JPO>
|
|
FMOVE.S #"$BF800000",FP1 ; odd, so invert and negate result <4/23/92, JPO>
|
|
FDIV.X FP0,FP1 ; fp1 = -1/tg(t)
|
|
FMOVE.X FP1,FP0 ; get ready to go: fp0 = -1/tg(t)
|
|
|
|
; still to do is to check for tg(t) = 0. if true then return infinity
|
|
; it does not seem to ever happen. what do we do?
|
|
;
|
|
; it does happen! take the π representation used in REM above and π/2 gives ∞.
|
|
; If tg was ±INF in last step by the nature of the algorithm, flipping the
|
|
; result sign corrects the error. 9/17/90 ... ali
|
|
|
|
BTST #10,D0 ; we already have fpsr, see if fp0 was 0
|
|
BEQ.S @1 ; if it is zero, then
|
|
FNEG.X FP0 ; flip the sign of INF in fp0
|
|
|
|
;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;
|
|
;; The algorithm for tg requires the computation of t^5, which will set the ;;
|
|
;; underflow flag when t is small enough ( in the order of 1.0e-1000 ) but ;;
|
|
;; not demormal. In that case we clear the underflow flag. 12/12/89...ali ;;
|
|
;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;
|
|
|
|
@1:
|
|
; BFEXTU (A1) {1:16},D0 ; get the exponent & 1st mantissa bit DELETED <4/23/92, JPO>
|
|
; MOVEA.L (SP)+,A1 ; restore the register after use DELETED <4/23/92, JPO>
|
|
; BEQ.B @2 ; if D0 = 0 then current s = denorm DELETED <4/23/92, JPO>
|
|
MOVE.L #SetInexact,D0 ; set the inexact flag
|
|
; MOVE.B #ClrUflow ,D0 ; and clear the underflow bit DELETED <4/23/92, JPO>
|
|
JMP (A4)
|
|
|
|
;@2 MOVE.L #SetInexact + SetUflow,D0 ; signal inexact DELETED <4/23/92, JPO>
|
|
; JMP (A4) ; deliver the result DELETED <4/23/92, JPO>
|
|
|
|
|
|
;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;
|
|
;; Problem cases: NaN, +∞ and zero. ;;
|
|
;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;
|
|
|
|
BadBoys BMI.B tgOut ; is arg = 0 ? ( N set )
|
|
BTST #29,D0 ; is arg a NaN? ( bit position 29 )
|
|
BNE.B tgOut ; scream, arg = NaN detected!
|
|
|
|
;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;
|
|
;; All right boys & girls, at this time it is apparent that the good old arg ;;
|
|
;; is an infinity or a NaN. ;;
|
|
;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;
|
|
|
|
FMOVE.X #NaNTrig,FP0 ; arg = +∞ => tg(arg) = NaN
|
|
MOVE.L #SetInvalid,D0 ; signal invalid
|
|
JMP (A4) ; bye
|
|
tgOut MOVEQ #0,D0 ; signal no exceptions
|
|
JMP (A4) ; return either NaN, +∞ or +-0 |