* * $VER: log.s 33.1 (20.1.97) * * " computes the natural logarithm of a normalized input" * * Version history: * * 33.1 20.1.97 (c) Motorola * * - cut'n pasted from M68060SP * * 33.2 22.1.97 (c) Motorola * * - added log10 and log2 * - log trashed a6, fixed * machine 68040 fpu 1 XDEF _log XDEF @log XDEF _log10 XDEF @log10 XDEF _log2 XDEF @log2 ************************************************************************* * log(): computes the natural logarithm of a normalized input * * * * INPUT *************************************************************** * * fp0 = extended precision input * * * * OUTPUT ************************************************************** * * fp0 = log(X) * * * * ACCURACY and MONOTONICITY ******************************************* * * The returned result is within 2 ulps 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. * * * * ALGORITHM *********************************************************** * * LOGN: * * Step 1. If |X-1| < 1/16, approximate log(X) by an odd * * polynomial in u, where u = 2(X-1)/(X+1). Otherwise, * * move on to Step 2. * * * * Step 2. X = 2**k * Y where 1 <= Y < 2. Define F to be the first * * seven significant bits of Y plus 2**(-7), i.e. * * F = 1.xxxxxx1 in base 2 where the six "x" match those * * of Y. Note that |Y-F| <= 2**(-7). * * * * Step 3. Define u = (Y-F)/F. Approximate log(1+u) by a * * polynomial in u, log(1+u) = poly. * * * * Step 4. Reconstruct * * log(X) = log( 2**k * Y ) = k*log(2) + log(F) + log(1+u) * * by k*log(2) + (log(F) + poly). The values of log(F) are * * calculated beforehand and stored in the program. * * * * Implementation Notes: * * Note 1. There are 64 different possible values for F, thus 64 * * log(F)'s need to be tabulated. Moreover, the values of * * 1/F are also tabulated so that the division in (Y-F)/F * * can be performed by a multiplication. * * * * Note 2. To fully exploit the pipeline, polynomials are usually * * separated into two parts evaluated independently before * * being added up. * * * ************************************************************************* cnop 0,4 LOGOF2 dc.l $3FFE0000,$B17217F7,$D1CF79AC,$00000000 one dc.l $3F800000 zero dc.l $00000000 infty dc.l $7F800000 negone dc.l $BF800000 LOGA6 dc.l $3FC2499A,$B5E4040B LOGA5 dc.l $BFC555B5,$848CB7DB LOGA4 dc.l $3FC99999,$987D8730 LOGA3 dc.l $BFCFFFFF,$FF6F7E97 LOGA2 dc.l $3FD55555,$555555A4 LOGA1 dc.l $BFE00000,$00000008 LOGB5 dc.l $3F175496,$ADD7DAD6 LOGB4 dc.l $3F3C71C2,$FE80C7E0 LOGB3 dc.l $3F624924,$928BCCFF LOGB2 dc.l $3F899999,$999995EC LOGB1 dc.l $3FB55555,$55555555 TWO dc.l $40000000,$00000000 LTHOLD dc.l $3f990000,$80000000,$00000000,$00000000 LOGTBL dc.l $3FFE0000,$FE03F80F,$E03F80FE,$00000000 dc.l $3FF70000,$FF015358,$833C47E2,$00000000 dc.l $3FFE0000,$FA232CF2,$52138AC0,$00000000 dc.l $3FF90000,$BDC8D83E,$AD88D549,$00000000 dc.l $3FFE0000,$F6603D98,$0F6603DA,$00000000 dc.l $3FFA0000,$9CF43DCF,$F5EAFD48,$00000000 dc.l $3FFE0000,$F2B9D648,$0F2B9D65,$00000000 dc.l $3FFA0000,$DA16EB88,$CB8DF614,$00000000 dc.l $3FFE0000,$EF2EB71F,$C4345238,$00000000 dc.l $3FFB0000,$8B29B775,$1BD70743,$00000000 dc.l $3FFE0000,$EBBDB2A5,$C1619C8C,$00000000 dc.l $3FFB0000,$A8D839F8,$30C1FB49,$00000000 dc.l $3FFE0000,$E865AC7B,$7603A197,$00000000 dc.l $3FFB0000,$C61A2EB1,$8CD907AD,$00000000 dc.l $3FFE0000,$E525982A,$F70C880E,$00000000 dc.l $3FFB0000,$E2F2A47A,$DE3A18AF,$00000000 dc.l $3FFE0000,$E1FC780E,$1FC780E2,$00000000 dc.l $3FFB0000,$FF64898E,$DF55D551,$00000000 dc.l $3FFE0000,$DEE95C4C,$A037BA57,$00000000 dc.l $3FFC0000,$8DB956A9,$7B3D0148,$00000000 dc.l $3FFE0000,$DBEB61EE,$D19C5958,$00000000 dc.l $3FFC0000,$9B8FE100,$F47BA1DE,$00000000 dc.l $3FFE0000,$D901B203,$6406C80E,$00000000 dc.l $3FFC0000,$A9372F1D,$0DA1BD17,$00000000 dc.l $3FFE0000,$D62B80D6,$2B80D62C,$00000000 dc.l $3FFC0000,$B6B07F38,$CE90E46B,$00000000 dc.l $3FFE0000,$D3680D36,$80D3680D,$00000000 dc.l $3FFC0000,$C3FD0329,$06488481,$00000000 dc.l $3FFE0000,$D0B69FCB,$D2580D0B,$00000000 dc.l $3FFC0000,$D11DE0FF,$15AB18CA,$00000000 dc.l $3FFE0000,$CE168A77,$25080CE1,$00000000 dc.l $3FFC0000,$DE1433A1,$6C66B150,$00000000 dc.l $3FFE0000,$CB8727C0,$65C393E0,$00000000 dc.l $3FFC0000,$EAE10B5A,$7DDC8ADD,$00000000 dc.l $3FFE0000,$C907DA4E,$871146AD,$00000000 dc.l $3FFC0000,$F7856E5E,$E2C9B291,$00000000 dc.l $3FFE0000,$C6980C69,$80C6980C,$00000000 dc.l $3FFD0000,$82012CA5,$A68206D7,$00000000 dc.l $3FFE0000,$C4372F85,$5D824CA6,$00000000 dc.l $3FFD0000,$882C5FCD,$7256A8C5,$00000000 dc.l $3FFE0000,$C1E4BBD5,$95F6E947,$00000000 dc.l $3FFD0000,$8E44C60B,$4CCFD7DE,$00000000 dc.l $3FFE0000,$BFA02FE8,$0BFA02FF,$00000000 dc.l $3FFD0000,$944AD09E,$F4351AF6,$00000000 dc.l $3FFE0000,$BD691047,$07661AA3,$00000000 dc.l $3FFD0000,$9A3EECD4,$C3EAA6B2,$00000000 dc.l $3FFE0000,$BB3EE721,$A54D880C,$00000000 dc.l $3FFD0000,$A0218434,$353F1DE8,$00000000 dc.l $3FFE0000,$B92143FA,$36F5E02E,$00000000 dc.l $3FFD0000,$A5F2FCAB,$BBC506DA,$00000000 dc.l $3FFE0000,$B70FBB5A,$19BE3659,$00000000 dc.l $3FFD0000,$ABB3B8BA,$2AD362A5,$00000000 dc.l $3FFE0000,$B509E68A,$9B94821F,$00000000 dc.l $3FFD0000,$B1641795,$CE3CA97B,$00000000 dc.l $3FFE0000,$B30F6352,$8917C80B,$00000000 dc.l $3FFD0000,$B7047551,$5D0F1C61,$00000000 dc.l $3FFE0000,$B11FD3B8,$0B11FD3C,$00000000 dc.l $3FFD0000,$BC952AFE,$EA3D13E1,$00000000 dc.l $3FFE0000,$AF3ADDC6,$80AF3ADE,$00000000 dc.l $3FFD0000,$C2168ED0,$F458BA4A,$00000000 dc.l $3FFE0000,$AD602B58,$0AD602B6,$00000000 dc.l $3FFD0000,$C788F439,$B3163BF1,$00000000 dc.l $3FFE0000,$AB8F69E2,$8359CD11,$00000000 dc.l $3FFD0000,$CCECAC08,$BF04565D,$00000000 dc.l $3FFE0000,$A9C84A47,$A07F5638,$00000000 dc.l $3FFD0000,$D2420487,$2DD85160,$00000000 dc.l $3FFE0000,$A80A80A8,$0A80A80B,$00000000 dc.l $3FFD0000,$D7894992,$3BC3588A,$00000000 dc.l $3FFE0000,$A655C439,$2D7B73A8,$00000000 dc.l $3FFD0000,$DCC2C4B4,$9887DACC,$00000000 dc.l $3FFE0000,$A4A9CF1D,$96833751,$00000000 dc.l $3FFD0000,$E1EEBD3E,$6D6A6B9E,$00000000 dc.l $3FFE0000,$A3065E3F,$AE7CD0E0,$00000000 dc.l $3FFD0000,$E70D785C,$2F9F5BDC,$00000000 dc.l $3FFE0000,$A16B312E,$A8FC377D,$00000000 dc.l $3FFD0000,$EC1F392C,$5179F283,$00000000 dc.l $3FFE0000,$9FD809FD,$809FD80A,$00000000 dc.l $3FFD0000,$F12440D3,$E36130E6,$00000000 dc.l $3FFE0000,$9E4CAD23,$DD5F3A20,$00000000 dc.l $3FFD0000,$F61CCE92,$346600BB,$00000000 dc.l $3FFE0000,$9CC8E160,$C3FB19B9,$00000000 dc.l $3FFD0000,$FB091FD3,$8145630A,$00000000 dc.l $3FFE0000,$9B4C6F9E,$F03A3CAA,$00000000 dc.l $3FFD0000,$FFE97042,$BFA4C2AD,$00000000 dc.l $3FFE0000,$99D722DA,$BDE58F06,$00000000 dc.l $3FFE0000,$825EFCED,$49369330,$00000000 dc.l $3FFE0000,$9868C809,$868C8098,$00000000 dc.l $3FFE0000,$84C37A7A,$B9A905C9,$00000000 dc.l $3FFE0000,$97012E02,$5C04B809,$00000000 dc.l $3FFE0000,$87224C2E,$8E645FB7,$00000000 dc.l $3FFE0000,$95A02568,$095A0257,$00000000 dc.l $3FFE0000,$897B8CAC,$9F7DE298,$00000000 dc.l $3FFE0000,$94458094,$45809446,$00000000 dc.l $3FFE0000,$8BCF55DE,$C4CD05FE,$00000000 dc.l $3FFE0000,$92F11384,$0497889C,$00000000 dc.l $3FFE0000,$8E1DC0FB,$89E125E5,$00000000 dc.l $3FFE0000,$91A2B3C4,$D5E6F809,$00000000 dc.l $3FFE0000,$9066E68C,$955B6C9B,$00000000 dc.l $3FFE0000,$905A3863,$3E06C43B,$00000000 dc.l $3FFE0000,$92AADE74,$C7BE59E0,$00000000 dc.l $3FFE0000,$8F1779D9,$FDC3A219,$00000000 dc.l $3FFE0000,$94E9BFF6,$15845643,$00000000 dc.l $3FFE0000,$8DDA5202,$37694809,$00000000 dc.l $3FFE0000,$9723A1B7,$20134203,$00000000 dc.l $3FFE0000,$8CA29C04,$6514E023,$00000000 dc.l $3FFE0000,$995899C8,$90EB8990,$00000000 dc.l $3FFE0000,$8B70344A,$139BC75A,$00000000 dc.l $3FFE0000,$9B88BDAA,$3A3DAE2F,$00000000 dc.l $3FFE0000,$8A42F870,$5669DB46,$00000000 dc.l $3FFE0000,$9DB4224F,$FFE1157C,$00000000 dc.l $3FFE0000,$891AC73A,$E9819B50,$00000000 dc.l $3FFE0000,$9FDADC26,$8B7A12DA,$00000000 dc.l $3FFE0000,$87F78087,$F78087F8,$00000000 dc.l $3FFE0000,$A1FCFF17,$CE733BD4,$00000000 dc.l $3FFE0000,$86D90544,$7A34ACC6,$00000000 dc.l $3FFE0000,$A41A9E8F,$5446FB9F,$00000000 dc.l $3FFE0000,$85BF3761,$2CEE3C9B,$00000000 dc.l $3FFE0000,$A633CD7E,$6771CD8B,$00000000 dc.l $3FFE0000,$84A9F9C8,$084A9F9D,$00000000 dc.l $3FFE0000,$A8489E60,$0B435A5E,$00000000 dc.l $3FFE0000,$83993052,$3FBE3368,$00000000 dc.l $3FFE0000,$AA59233C,$CCA4BD49,$00000000 dc.l $3FFE0000,$828CBFBE,$B9A020A3,$00000000 dc.l $3FFE0000,$AC656DAE,$6BCC4985,$00000000 dc.l $3FFE0000,$81848DA8,$FAF0D277,$00000000 dc.l $3FFE0000,$AE6D8EE3,$60BB2468,$00000000 dc.l $3FFE0000,$80808080,$80808081,$00000000 dc.l $3FFE0000,$B07197A2,$3C46C654,$00000000 X EQU -12 XDCARE EQU X+2 XFRAC EQU X+4 KLOG2 EQU -12 SAVEU EQU -12 F EQU -24 FFRAC EQU F+4 TEMP_SIZE EQU 24 _log fmove.d (4,sp),fp0 @log link a1,#-TEMP_SIZE fmove.x fp0,(X,a1) .LOGBGN ;--FPCR SAVED AND CLEARED, INPUT IS 2^(ADJK)*FP0, FP0 CONTAINS ;--A FINITE, NON-ZERO, NORMALIZED NUMBER. move.l (X,a1),d1 move.w (X+4,a1),d1 tst.l d1 ; CHECK IF X IS NEGATIVE ble.w .LOGNEG ; LOG OF NEGATIVE ARGUMENT IS INVALID ; X IS POSITIVE, CHECK IF X IS NEAR 1 cmp.l #$3ffef07d,d1 ; IS X < 15/16? blt.b .LOGMAIN ; YES cmp.l #$3fff8841,d1 ; IS X > 17/16? ble.w .LOGNEAR1 ; NO .LOGMAIN ;--THIS SHOULD BE THE USUAL CASE, X NOT VERY CLOSE TO 1 ;--X = 2^(K) * Y, 1 <= Y < 2. THUS, Y = 1.XXXXXXXX....XX IN BINARY. ;--WE DEFINE F = 1.XXXXXX1, I.E. FIRST 7 BITS OF Y AND ATTACH A 1. ;--THE IDEA IS THAT LOG(X) = K*LOG2 + LOG(Y) ;-- = K*LOG2 + LOG(F) + LOG(1 + (Y-F)/F). ;--NOTE THAT U = (Y-F)/F IS VERY SMALL AND THUS APPROXIMATING ;--LOG(1+U) CAN BE VERY EFFICIENT. ;--ALSO NOTE THAT THE VALUE 1/F IS STORED IN A TABLE SO THAT NO ;--DIVISION IS NEEDED TO CALCULATE (Y-F)/F. ;--GET K, Y, F, AND ADDRESS OF 1/F. asr.l #8,d1 asr.l #8,d1 ; SHIFTED 16 BITS, BIASED EXPO. OF X sub.l #$3FFF,d1 ; THIS IS K lea (LOGTBL,pc),a0 ; BASE ADDRESS OF 1/F AND LOG(F) fmove.l d1,fp1 ; CONVERT K TO FLOATING-POINT FORMAT ;--WHILE THE CONVERSION IS GOING ON, WE GET F AND ADDRESS OF 1/F move.l #$3FFF0000,(X,a1) ; X IS NOW Y, I.E. 2^(-K)*X move.l (XFRAC,a1),(FFRAC,a1) and.l #$FE000000,(FFRAC,a1) ; FIRST 7 BITS OF Y or.l #$01000000,(FFRAC,a1) move.l (FFRAC,a1),d1 and.l #$7E000000,d1 asr.l #8,d1 asr.l #8,d1 asr.l #4,d1 ; SHIFTED 20, D0 IS THE DISPLACEMENT add.l d1,a0 ; A0 IS THE ADDRESS FOR 1/F fmove.x (X,a1),fp0 move.l #$3fff0000,(F,a1) clr.l (F+8,a1) fsub.x (F,a1),fp0 ; Y-F fmovem.x fp2/fp3,-(sp) ; SAVE FP2-3 WHILE FP0 IS NOT READY ;--SUMMARY: FP0 IS Y-F, A0 IS ADDRESS OF 1/F, FP1 IS K ;--REGISTERS SAVED: FPCR, FP1, FP2 .LP1CONT1 ;--AN RE-ENTRY POINT FOR LOGNP1 fmul.x (a0),fp0 ; FP0 IS U = (Y-F)/F fmul.x (LOGOF2,pc),fp1 ; GET K*LOG2 WHILE FP0 IS NOT READY fmove.x fp0,fp2 fmul.x fp2,fp2 ; FP2 IS V=U*U fmove.x fp1,(KLOG2,a1) ; PUT K*LOG2 IN MEMEORY, FREE FP1 ;--LOG(1+U) IS APPROXIMATED BY ;--U + V*(A1+U*(A2+U*(A3+U*(A4+U*(A5+U*A6))))) WHICH IS ;--[U + V*(A1+V*(A3+V*A5))] + [U*V*(A2+V*(A4+V*A6))] fmove.x fp2,fp3 fmove.x fp2,fp1 fmul.d (LOGA6,pc),fp1 ; V*A6 fmul.d (LOGA5,pc),fp2 ; V*A5 fadd.d (LOGA4,pc),fp1 ; A4+V*A6 fadd.d (LOGA3,pc),fp2 ; A3+V*A5 fmul.x fp3,fp1 ; V*(A4+V*A6) fmul.x fp3,fp2 ; V*(A3+V*A5) fadd.d (LOGA2,pc),fp1 ; A2+V*(A4+V*A6) fadd.d (LOGA1,pc),fp2 ; A1+V*(A3+V*A5) fmul.x fp3,fp1 ; V*(A2+V*(A4+V*A6)) addq.l #8,a0 ; ADDRESS OF LOG(F) fmul.x fp3,fp2 ; V*(A1+V*(A3+V*A5)) addq.l #8,a0 fmul.x fp0,fp1 ; U*V*(A2+V*(A4+V*A6)) fadd.x fp2,fp0 ; U+V*(A1+V*(A3+V*A5)) fadd.x (a0),fp1 ; LOG(F)+U*V*(A2+V*(A4+V*A6)) fmovem.x (sp)+,fp2/fp3 ; RESTORE FP2-3 fadd.x fp1,fp0 ; FP0 IS LOG(F) + LOG(1+U) fadd.x (KLOG2,a1),fp0 ; FINAL ADD unlk a1 rts .LOGNEAR1 ; if the input is exactly equal to one, then exit through ld_pzero. ; if these 2 lines weren't here, the correct answer would be returned ; but the INEX2 bit would be set. fcmp.s #1,fp0 ; is it equal to one? fbeq .pzero ; yes ;--REGISTERS SAVED: FPCR, FP1. FP0 CONTAINS THE INPUT. fmove.x fp0,fp1 fsub.s (one,pc),fp1 ; FP1 IS X-1 fadd.s (one,pc),fp0 ; FP0 IS X+1 fadd.x fp1,fp1 ; FP1 IS 2(X-1) ;--LOG(X) = LOG(1+U/2)-LOG(1-U/2) WHICH IS AN ODD POLYNOMIAL ;--IN U, U = 2(X-1)/(X+1) = FP1/FP0 .LP1CONT2 ;--THIS IS AN RE-ENTRY POINT FOR LOGNP1 fdiv.x fp0,fp1 ; FP1 IS U fmovem.x fp2/fp3,-(sp) ; SAVE FP2-3 ;--REGISTERS SAVED ARE NOW FPCR,FP1,FP2,FP3 ;--LET V=U*U, W=V*V, CALCULATE ;--U + U*V*(B1 + V*(B2 + V*(B3 + V*(B4 + V*B5)))) BY ;--U + U*V*( [B1 + W*(B3 + W*B5)] + [V*(B2 + W*B4)] ) fmove.x fp1,fp0 fmul.x fp0,fp0 ; FP0 IS V fmove.x fp1,(SAVEU,a1) ; STORE U IN MEMORY, FREE FP1 fmove.x fp0,fp1 fmul.x fp1,fp1 ; FP1 IS W fmove.d (LOGB5,pc),fp3 fmove.d (LOGB4,pc),fp2 fmul.x fp1,fp3 ; W*B5 fmul.x fp1,fp2 ; W*B4 fadd.d (LOGB3,pc),fp3 ; B3+W*B5 fadd.d (LOGB2,pc),fp2 ; B2+W*B4 fmul.x fp3,fp1 ; W*(B3+W*B5), FP3 RELEASED fmul.x fp0,fp2 ; V*(B2+W*B4) fadd.d (LOGB1,pc),fp1 ; B1+W*(B3+W*B5) fmul.x (SAVEU,a1),fp0 ; FP0 IS U*V fadd.x fp2,fp1 ; B1+W*(B3+W*B5) + V*(B2+W*B4), FP2 RELEASED fmovem.x (sp)+,fp2/fp3 ; FP2-3 RESTORED fmul.x fp1,fp0 ; U*V*( [B1+W*(B3+W*B5)] + [V*(B2+W*B4)] ) fadd.x (SAVEU,a1),fp0 unlk a1 rts ;--REGISTERS SAVED FPCR. LOG(-VE) IS INVALID .pzero fmove.s #0,fp0 .LOGNEG unlk a1 rts ************************************************************************* * log10(): computes the base-10 logarithm of a normalized input * * log2(): computes the base-2 logarithm of a normalized input * * * * INPUT *************************************************************** * * fp0 = extended precision input * * * * OUTPUT ************************************************************** * * fp0 = log10(X) or log2(X) * * * * ACCURACY and MONOTONICITY ******************************************* * * The returned result is within 1.7 ulps in 64 significant bit, * * i.e. within 0.5003 ulp to 53 bits if the result is subsequently * * rounded to double precision. The result is provably monotonic * * in double precision. * * * * ALGORITHM *********************************************************** * * * * log10: * * * * Step 0. If X < 0, create a NaN and raise the invalid operation * * flag. Otherwise, save FPCR in D1; set FpCR to default. * * Notes: Default means round-to-nearest mode, no floating-point * * traps, and precision control = double extended. * * * * Step 1. Call sLogN to obtain Y = log(X), the natural log of X. * * * * Step 2. Compute log_10(X) = log(X) * (1/log(10)). * * 2.1 Restore the user FPCR * * 2.2 Return ans := Y * INV_L10. * * * * log2: * * * * Step 0. If X < 0, create a NaN and raise the invalid operation * * flag. Otherwise, save FPCR in D1; set FpCR to default. * * Notes: Default means round-to-nearest mode, no floating-point * * traps, and precision control = double extended. * * * * Step 1. If X is not an integer power of two, i.e., X != 2^k, * * go to Step 3. * * * * Step 2. Return k. * * 2.1 Get integer k, X = 2^k. * * 2.2 Restore the user FPCR. * * 2.3 Return ans := convert-to-double-extended(k). * * * * Step 3. Call sLogN to obtain Y = log(X), the natural log of X. * * * * Step 4. Compute log_2(X) = log(X) * (1/log(2)). * * 4.1 Restore the user FPCR * * 4.2 Return ans := Y * INV_L2. * * * ************************************************************************* cnop 0,8 INV_L10 dc.l $3FFD0000,$DE5BD8A9,$37287195,$00000000 INV_L2 dc.l $3FFF0000,$B8AA3B29,$5C17F0BC,$00000000 *--entry point for log10(X), X is normalized _log10 fmove.d (4,sp),fp0 @log10 fmove.s fp0,-(sp) tst.l (sp) blt.s .invalid fcmp.s #1,fp0 fbeq .zero addq.l #4,sp bsr.w @log ; log(X), X normal. fmul.x (INV_L10,pc),fp0 rts .zero fmove.s #0,fp0 .invalid addq.l #4,sp rts ;--entry point for Log2(X), X is normalized _log2 fmove.d (4,sp),fp0 @log2 fmove.x fp0,-(sp) tst.l (sp) blt.s .invalid tst.l (8,sp) bne .continue move.l (4,sp),d1 and.l #$7FFFFFFF,d1 bne.b .continue ; X is not 2^k ;--X = 2^k. move.w (sp),d1 and.l #$00007FFF,d1 lea (12,sp),sp sub.l #$3FFF,d1 beq.l .zero fmove.l d1,fp0 rts .zero fmove.s #0,fp0 rts .continue: bsr @log ; log(X), X normal. fmul.x (INV_L2,pc),fp0 .invalid lea (12,sp),sp rts