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+|
+| stan.sa 3.3 7/29/91
+|
+| The entry point stan computes the tangent of
+| an input argument;
+| stand does the same except for denormalized input.
+|
+| Input: Double-extended number X in location pointed to
+| by address register a0.
+|
+| Output: The value tan(X) returned in floating-point register Fp0.
+|
+| Accuracy and Monotonicity: The returned result is within 3 ulp 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 sTAN takes approximately 170 cycles for
+| input argument X such that |X| < 15Pi, which is the usual
+| situation.
+|
+| Algorithm:
+|
+| 1. If |X| >= 15Pi or |X| < 2**(-40), go to 6.
+|
+| 2. Decompose X as X = N(Pi/2) + r where |r| <= Pi/4. Let
+| k = N mod 2, so in particular, k = 0 or 1.
+|
+| 3. If k is odd, go to 5.
+|
+| 4. (k is even) Tan(X) = tan(r) and tan(r) is approximated by a
+| rational function U/V where
+| U = r + r*s*(P1 + s*(P2 + s*P3)), and
+| V = 1 + s*(Q1 + s*(Q2 + s*(Q3 + s*Q4))), s = r*r.
+| Exit.
+|
+| 4. (k is odd) Tan(X) = -cot(r). Since tan(r) is approximated by a
+| rational function U/V where
+| U = r + r*s*(P1 + s*(P2 + s*P3)), and
+| V = 1 + s*(Q1 + s*(Q2 + s*(Q3 + s*Q4))), s = r*r,
+| -Cot(r) = -V/U. Exit.
+|
+| 6. If |X| > 1, go to 8.
+|
+| 7. (|X|<2**(-40)) Tan(X) = X. Exit.
+|
+| 8. Overwrite X by X := X rem 2Pi. Now that |X| <= Pi, go back to 2.
+|
+
+| Copyright (C) Motorola, Inc. 1990
+| All Rights Reserved
+|
+| For details on the license for this file, please see the
+| file, README, in this same directory.
+
+|STAN idnt 2,1 | Motorola 040 Floating Point Software Package
+
+ |section 8
+
+#include "fpsp.h"
+
+BOUNDS1: .long 0x3FD78000,0x4004BC7E
+TWOBYPI: .long 0x3FE45F30,0x6DC9C883
+
+TANQ4: .long 0x3EA0B759,0xF50F8688
+TANP3: .long 0xBEF2BAA5,0xA8924F04
+
+TANQ3: .long 0xBF346F59,0xB39BA65F,0x00000000,0x00000000
+
+TANP2: .long 0x3FF60000,0xE073D3FC,0x199C4A00,0x00000000
+
+TANQ2: .long 0x3FF90000,0xD23CD684,0x15D95FA1,0x00000000
+
+TANP1: .long 0xBFFC0000,0x8895A6C5,0xFB423BCA,0x00000000
+
+TANQ1: .long 0xBFFD0000,0xEEF57E0D,0xA84BC8CE,0x00000000
+
+INVTWOPI: .long 0x3FFC0000,0xA2F9836E,0x4E44152A,0x00000000
+
+TWOPI1: .long 0x40010000,0xC90FDAA2,0x00000000,0x00000000
+TWOPI2: .long 0x3FDF0000,0x85A308D4,0x00000000,0x00000000
+
+|--N*PI/2, -32 <= N <= 32, IN A LEADING TERM IN EXT. AND TRAILING
+|--TERM IN SGL. NOTE THAT PI IS 64-BIT LONG, THUS N*PI/2 IS AT
+|--MOST 69 BITS LONG.
+ .global PITBL
+PITBL:
+ .long 0xC0040000,0xC90FDAA2,0x2168C235,0x21800000
+ .long 0xC0040000,0xC2C75BCD,0x105D7C23,0xA0D00000
+ .long 0xC0040000,0xBC7EDCF7,0xFF523611,0xA1E80000
+ .long 0xC0040000,0xB6365E22,0xEE46F000,0x21480000
+ .long 0xC0040000,0xAFEDDF4D,0xDD3BA9EE,0xA1200000
+ .long 0xC0040000,0xA9A56078,0xCC3063DD,0x21FC0000
+ .long 0xC0040000,0xA35CE1A3,0xBB251DCB,0x21100000
+ .long 0xC0040000,0x9D1462CE,0xAA19D7B9,0xA1580000
+ .long 0xC0040000,0x96CBE3F9,0x990E91A8,0x21E00000
+ .long 0xC0040000,0x90836524,0x88034B96,0x20B00000
+ .long 0xC0040000,0x8A3AE64F,0x76F80584,0xA1880000
+ .long 0xC0040000,0x83F2677A,0x65ECBF73,0x21C40000
+ .long 0xC0030000,0xFB53D14A,0xA9C2F2C2,0x20000000
+ .long 0xC0030000,0xEEC2D3A0,0x87AC669F,0x21380000
+ .long 0xC0030000,0xE231D5F6,0x6595DA7B,0xA1300000
+ .long 0xC0030000,0xD5A0D84C,0x437F4E58,0x9FC00000
+ .long 0xC0030000,0xC90FDAA2,0x2168C235,0x21000000
+ .long 0xC0030000,0xBC7EDCF7,0xFF523611,0xA1680000
+ .long 0xC0030000,0xAFEDDF4D,0xDD3BA9EE,0xA0A00000
+ .long 0xC0030000,0xA35CE1A3,0xBB251DCB,0x20900000
+ .long 0xC0030000,0x96CBE3F9,0x990E91A8,0x21600000
+ .long 0xC0030000,0x8A3AE64F,0x76F80584,0xA1080000
+ .long 0xC0020000,0xFB53D14A,0xA9C2F2C2,0x1F800000
+ .long 0xC0020000,0xE231D5F6,0x6595DA7B,0xA0B00000
+ .long 0xC0020000,0xC90FDAA2,0x2168C235,0x20800000
+ .long 0xC0020000,0xAFEDDF4D,0xDD3BA9EE,0xA0200000
+ .long 0xC0020000,0x96CBE3F9,0x990E91A8,0x20E00000
+ .long 0xC0010000,0xFB53D14A,0xA9C2F2C2,0x1F000000
+ .long 0xC0010000,0xC90FDAA2,0x2168C235,0x20000000
+ .long 0xC0010000,0x96CBE3F9,0x990E91A8,0x20600000
+ .long 0xC0000000,0xC90FDAA2,0x2168C235,0x1F800000
+ .long 0xBFFF0000,0xC90FDAA2,0x2168C235,0x1F000000
+ .long 0x00000000,0x00000000,0x00000000,0x00000000
+ .long 0x3FFF0000,0xC90FDAA2,0x2168C235,0x9F000000
+ .long 0x40000000,0xC90FDAA2,0x2168C235,0x9F800000
+ .long 0x40010000,0x96CBE3F9,0x990E91A8,0xA0600000
+ .long 0x40010000,0xC90FDAA2,0x2168C235,0xA0000000
+ .long 0x40010000,0xFB53D14A,0xA9C2F2C2,0x9F000000
+ .long 0x40020000,0x96CBE3F9,0x990E91A8,0xA0E00000
+ .long 0x40020000,0xAFEDDF4D,0xDD3BA9EE,0x20200000
+ .long 0x40020000,0xC90FDAA2,0x2168C235,0xA0800000
+ .long 0x40020000,0xE231D5F6,0x6595DA7B,0x20B00000
+ .long 0x40020000,0xFB53D14A,0xA9C2F2C2,0x9F800000
+ .long 0x40030000,0x8A3AE64F,0x76F80584,0x21080000
+ .long 0x40030000,0x96CBE3F9,0x990E91A8,0xA1600000
+ .long 0x40030000,0xA35CE1A3,0xBB251DCB,0xA0900000
+ .long 0x40030000,0xAFEDDF4D,0xDD3BA9EE,0x20A00000
+ .long 0x40030000,0xBC7EDCF7,0xFF523611,0x21680000
+ .long 0x40030000,0xC90FDAA2,0x2168C235,0xA1000000
+ .long 0x40030000,0xD5A0D84C,0x437F4E58,0x1FC00000
+ .long 0x40030000,0xE231D5F6,0x6595DA7B,0x21300000
+ .long 0x40030000,0xEEC2D3A0,0x87AC669F,0xA1380000
+ .long 0x40030000,0xFB53D14A,0xA9C2F2C2,0xA0000000
+ .long 0x40040000,0x83F2677A,0x65ECBF73,0xA1C40000
+ .long 0x40040000,0x8A3AE64F,0x76F80584,0x21880000
+ .long 0x40040000,0x90836524,0x88034B96,0xA0B00000
+ .long 0x40040000,0x96CBE3F9,0x990E91A8,0xA1E00000
+ .long 0x40040000,0x9D1462CE,0xAA19D7B9,0x21580000
+ .long 0x40040000,0xA35CE1A3,0xBB251DCB,0xA1100000
+ .long 0x40040000,0xA9A56078,0xCC3063DD,0xA1FC0000
+ .long 0x40040000,0xAFEDDF4D,0xDD3BA9EE,0x21200000
+ .long 0x40040000,0xB6365E22,0xEE46F000,0xA1480000
+ .long 0x40040000,0xBC7EDCF7,0xFF523611,0x21E80000
+ .long 0x40040000,0xC2C75BCD,0x105D7C23,0x20D00000
+ .long 0x40040000,0xC90FDAA2,0x2168C235,0xA1800000
+
+ .set INARG,FP_SCR4
+
+ .set TWOTO63,L_SCR1
+ .set ENDFLAG,L_SCR2
+ .set N,L_SCR3
+
+ | xref t_frcinx
+ |xref t_extdnrm
+
+ .global stand
+stand:
+|--TAN(X) = X FOR DENORMALIZED X
+
+ bra t_extdnrm
+
+ .global stan
+stan:
+ fmovex (%a0),%fp0 | ...LOAD INPUT
+
+ movel (%a0),%d0
+ movew 4(%a0),%d0
+ andil #0x7FFFFFFF,%d0
+
+ cmpil #0x3FD78000,%d0 | ...|X| >= 2**(-40)?
+ bges TANOK1
+ bra TANSM
+TANOK1:
+ cmpil #0x4004BC7E,%d0 | ...|X| < 15 PI?
+ blts TANMAIN
+ bra REDUCEX
+
+
+TANMAIN:
+|--THIS IS THE USUAL CASE, |X| <= 15 PI.
+|--THE ARGUMENT REDUCTION IS DONE BY TABLE LOOK UP.
+ fmovex %fp0,%fp1
+ fmuld TWOBYPI,%fp1 | ...X*2/PI
+
+|--HIDE THE NEXT TWO INSTRUCTIONS
+ leal PITBL+0x200,%a1 | ...TABLE OF N*PI/2, N = -32,...,32
+
+|--FP1 IS NOW READY
+ fmovel %fp1,%d0 | ...CONVERT TO INTEGER
+
+ asll #4,%d0
+ addal %d0,%a1 | ...ADDRESS N*PIBY2 IN Y1, Y2
+
+ fsubx (%a1)+,%fp0 | ...X-Y1
+|--HIDE THE NEXT ONE
+
+ fsubs (%a1),%fp0 | ...FP0 IS R = (X-Y1)-Y2
+
+ rorl #5,%d0
+ andil #0x80000000,%d0 | ...D0 WAS ODD IFF D0 < 0
+
+TANCONT:
+
+ cmpil #0,%d0
+ blt NODD
+
+ fmovex %fp0,%fp1
+ fmulx %fp1,%fp1 | ...S = R*R
+
+ fmoved TANQ4,%fp3
+ fmoved TANP3,%fp2
+
+ fmulx %fp1,%fp3 | ...SQ4
+ fmulx %fp1,%fp2 | ...SP3
+
+ faddd TANQ3,%fp3 | ...Q3+SQ4
+ faddx TANP2,%fp2 | ...P2+SP3
+
+ fmulx %fp1,%fp3 | ...S(Q3+SQ4)
+ fmulx %fp1,%fp2 | ...S(P2+SP3)
+
+ faddx TANQ2,%fp3 | ...Q2+S(Q3+SQ4)
+ faddx TANP1,%fp2 | ...P1+S(P2+SP3)
+
+ fmulx %fp1,%fp3 | ...S(Q2+S(Q3+SQ4))
+ fmulx %fp1,%fp2 | ...S(P1+S(P2+SP3))
+
+ faddx TANQ1,%fp3 | ...Q1+S(Q2+S(Q3+SQ4))
+ fmulx %fp0,%fp2 | ...RS(P1+S(P2+SP3))
+
+ fmulx %fp3,%fp1 | ...S(Q1+S(Q2+S(Q3+SQ4)))
+
+
+ faddx %fp2,%fp0 | ...R+RS(P1+S(P2+SP3))
+
+
+ fadds #0x3F800000,%fp1 | ...1+S(Q1+...)
+
+ fmovel %d1,%fpcr |restore users exceptions
+ fdivx %fp1,%fp0 |last inst - possible exception set
+
+ bra t_frcinx
+
+NODD:
+ fmovex %fp0,%fp1
+ fmulx %fp0,%fp0 | ...S = R*R
+
+ fmoved TANQ4,%fp3
+ fmoved TANP3,%fp2
+
+ fmulx %fp0,%fp3 | ...SQ4
+ fmulx %fp0,%fp2 | ...SP3
+
+ faddd TANQ3,%fp3 | ...Q3+SQ4
+ faddx TANP2,%fp2 | ...P2+SP3
+
+ fmulx %fp0,%fp3 | ...S(Q3+SQ4)
+ fmulx %fp0,%fp2 | ...S(P2+SP3)
+
+ faddx TANQ2,%fp3 | ...Q2+S(Q3+SQ4)
+ faddx TANP1,%fp2 | ...P1+S(P2+SP3)
+
+ fmulx %fp0,%fp3 | ...S(Q2+S(Q3+SQ4))
+ fmulx %fp0,%fp2 | ...S(P1+S(P2+SP3))
+
+ faddx TANQ1,%fp3 | ...Q1+S(Q2+S(Q3+SQ4))
+ fmulx %fp1,%fp2 | ...RS(P1+S(P2+SP3))
+
+ fmulx %fp3,%fp0 | ...S(Q1+S(Q2+S(Q3+SQ4)))
+
+
+ faddx %fp2,%fp1 | ...R+RS(P1+S(P2+SP3))
+ fadds #0x3F800000,%fp0 | ...1+S(Q1+...)
+
+
+ fmovex %fp1,-(%sp)
+ eoril #0x80000000,(%sp)
+
+ fmovel %d1,%fpcr |restore users exceptions
+ fdivx (%sp)+,%fp0 |last inst - possible exception set
+
+ bra t_frcinx
+
+TANBORS:
+|--IF |X| > 15PI, WE USE THE GENERAL ARGUMENT REDUCTION.
+|--IF |X| < 2**(-40), RETURN X OR 1.
+ cmpil #0x3FFF8000,%d0
+ bgts REDUCEX
+
+TANSM:
+
+ fmovex %fp0,-(%sp)
+ fmovel %d1,%fpcr |restore users exceptions
+ fmovex (%sp)+,%fp0 |last inst - possible exception set
+
+ bra t_frcinx
+
+
+REDUCEX:
+|--WHEN REDUCEX IS USED, THE CODE WILL INEVITABLY BE SLOW.
+|--THIS REDUCTION METHOD, HOWEVER, IS MUCH FASTER THAN USING
+|--THE REMAINDER INSTRUCTION WHICH IS NOW IN SOFTWARE.
+
+ fmovemx %fp2-%fp5,-(%a7) | ...save FP2 through FP5
+ movel %d2,-(%a7)
+ fmoves #0x00000000,%fp1
+
+|--If compact form of abs(arg) in d0=$7ffeffff, argument is so large that
+|--there is a danger of unwanted overflow in first LOOP iteration. In this
+|--case, reduce argument by one remainder step to make subsequent reduction
+|--safe.
+ cmpil #0x7ffeffff,%d0 |is argument dangerously large?
+ bnes LOOP
+ movel #0x7ffe0000,FP_SCR2(%a6) |yes
+| ;create 2**16383*PI/2
+ movel #0xc90fdaa2,FP_SCR2+4(%a6)
+ clrl FP_SCR2+8(%a6)
+ ftstx %fp0 |test sign of argument
+ movel #0x7fdc0000,FP_SCR3(%a6) |create low half of 2**16383*
+| ;PI/2 at FP_SCR3
+ movel #0x85a308d3,FP_SCR3+4(%a6)
+ clrl FP_SCR3+8(%a6)
+ fblt red_neg
+ orw #0x8000,FP_SCR2(%a6) |positive arg
+ orw #0x8000,FP_SCR3(%a6)
+red_neg:
+ faddx FP_SCR2(%a6),%fp0 |high part of reduction is exact
+ fmovex %fp0,%fp1 |save high result in fp1
+ faddx FP_SCR3(%a6),%fp0 |low part of reduction
+ fsubx %fp0,%fp1 |determine low component of result
+ faddx FP_SCR3(%a6),%fp1 |fp0/fp1 are reduced argument.
+
+|--ON ENTRY, FP0 IS X, ON RETURN, FP0 IS X REM PI/2, |X| <= PI/4.
+|--integer quotient will be stored in N
+|--Intermediate remainder is 66-bit long; (R,r) in (FP0,FP1)
+
+LOOP:
+ fmovex %fp0,INARG(%a6) | ...+-2**K * F, 1 <= F < 2
+ movew INARG(%a6),%d0
+ movel %d0,%a1 | ...save a copy of D0
+ andil #0x00007FFF,%d0
+ subil #0x00003FFF,%d0 | ...D0 IS K
+ cmpil #28,%d0
+ bles LASTLOOP
+CONTLOOP:
+ subil #27,%d0 | ...D0 IS L := K-27
+ movel #0,ENDFLAG(%a6)
+ bras WORK
+LASTLOOP:
+ clrl %d0 | ...D0 IS L := 0
+ movel #1,ENDFLAG(%a6)
+
+WORK:
+|--FIND THE REMAINDER OF (R,r) W.R.T. 2**L * (PI/2). L IS SO CHOSEN
+|--THAT INT( X * (2/PI) / 2**(L) ) < 2**29.
+
+|--CREATE 2**(-L) * (2/PI), SIGN(INARG)*2**(63),
+|--2**L * (PIby2_1), 2**L * (PIby2_2)
+
+ movel #0x00003FFE,%d2 | ...BIASED EXPO OF 2/PI
+ subl %d0,%d2 | ...BIASED EXPO OF 2**(-L)*(2/PI)
+
+ movel #0xA2F9836E,FP_SCR1+4(%a6)
+ movel #0x4E44152A,FP_SCR1+8(%a6)
+ movew %d2,FP_SCR1(%a6) | ...FP_SCR1 is 2**(-L)*(2/PI)
+
+ fmovex %fp0,%fp2
+ fmulx FP_SCR1(%a6),%fp2
+|--WE MUST NOW FIND INT(FP2). SINCE WE NEED THIS VALUE IN
+|--FLOATING POINT FORMAT, THE TWO FMOVE'S FMOVE.L FP <--> N
+|--WILL BE TOO INEFFICIENT. THE WAY AROUND IT IS THAT
+|--(SIGN(INARG)*2**63 + FP2) - SIGN(INARG)*2**63 WILL GIVE
+|--US THE DESIRED VALUE IN FLOATING POINT.
+
+|--HIDE SIX CYCLES OF INSTRUCTION
+ movel %a1,%d2
+ swap %d2
+ andil #0x80000000,%d2
+ oril #0x5F000000,%d2 | ...D2 IS SIGN(INARG)*2**63 IN SGL
+ movel %d2,TWOTO63(%a6)
+
+ movel %d0,%d2
+ addil #0x00003FFF,%d2 | ...BIASED EXPO OF 2**L * (PI/2)
+
+|--FP2 IS READY
+ fadds TWOTO63(%a6),%fp2 | ...THE FRACTIONAL PART OF FP1 IS ROUNDED
+
+|--HIDE 4 CYCLES OF INSTRUCTION; creating 2**(L)*Piby2_1 and 2**(L)*Piby2_2
+ movew %d2,FP_SCR2(%a6)
+ clrw FP_SCR2+2(%a6)
+ movel #0xC90FDAA2,FP_SCR2+4(%a6)
+ clrl FP_SCR2+8(%a6) | ...FP_SCR2 is 2**(L) * Piby2_1
+
+|--FP2 IS READY
+ fsubs TWOTO63(%a6),%fp2 | ...FP2 is N
+
+ addil #0x00003FDD,%d0
+ movew %d0,FP_SCR3(%a6)
+ clrw FP_SCR3+2(%a6)
+ movel #0x85A308D3,FP_SCR3+4(%a6)
+ clrl FP_SCR3+8(%a6) | ...FP_SCR3 is 2**(L) * Piby2_2
+
+ movel ENDFLAG(%a6),%d0
+
+|--We are now ready to perform (R+r) - N*P1 - N*P2, P1 = 2**(L) * Piby2_1 and
+|--P2 = 2**(L) * Piby2_2
+ fmovex %fp2,%fp4
+ fmulx FP_SCR2(%a6),%fp4 | ...W = N*P1
+ fmovex %fp2,%fp5
+ fmulx FP_SCR3(%a6),%fp5 | ...w = N*P2
+ fmovex %fp4,%fp3
+|--we want P+p = W+w but |p| <= half ulp of P
+|--Then, we need to compute A := R-P and a := r-p
+ faddx %fp5,%fp3 | ...FP3 is P
+ fsubx %fp3,%fp4 | ...W-P
+
+ fsubx %fp3,%fp0 | ...FP0 is A := R - P
+ faddx %fp5,%fp4 | ...FP4 is p = (W-P)+w
+
+ fmovex %fp0,%fp3 | ...FP3 A
+ fsubx %fp4,%fp1 | ...FP1 is a := r - p
+
+|--Now we need to normalize (A,a) to "new (R,r)" where R+r = A+a but
+|--|r| <= half ulp of R.
+ faddx %fp1,%fp0 | ...FP0 is R := A+a
+|--No need to calculate r if this is the last loop
+ cmpil #0,%d0
+ bgt RESTORE
+
+|--Need to calculate r
+ fsubx %fp0,%fp3 | ...A-R
+ faddx %fp3,%fp1 | ...FP1 is r := (A-R)+a
+ bra LOOP
+
+RESTORE:
+ fmovel %fp2,N(%a6)
+ movel (%a7)+,%d2
+ fmovemx (%a7)+,%fp2-%fp5
+
+
+ movel N(%a6),%d0
+ rorl #1,%d0
+
+
+ bra TANCONT
+
+ |end
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