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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
|
|       THIS IS UNPUBLISHED PROPRIETARY SOURCE CODE OF MOTOROLA 
|       The copyright notice above does not evidence any  
|       actual or intended publication of such source code.

|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