blob: f8553aaececbc64d76be7908e57b43869089558e [file] [log] [blame]
 | | 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