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| -rw-r--r-- | arch/m68k/fpsp040/stan.S | 454 |
1 files changed, 454 insertions, 0 deletions
diff --git a/arch/m68k/fpsp040/stan.S b/arch/m68k/fpsp040/stan.S new file mode 100644 index 000000000..f8553aaec --- /dev/null +++ b/arch/m68k/fpsp040/stan.S @@ -0,0 +1,454 @@ +| +| 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 |