864 lines
30 KiB
NASM
864 lines
30 KiB
NASM
subttl emtrig.asm - Trig functions sine, cosine, tangent
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page
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; Copyright (c) Microsoft Corporation 1991
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; All Rights Reserved
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;emtrig.asm - Trig functions sine, cosine, tangent
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; by Tim Paterson
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;Purpose:
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; FCOS, FPTAN, FSIN, FSINCOS instructions
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;Inputs:
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; edi = [CURstk]
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;Revision History:
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; [] 09/05/91 TP Initial 32-bit version.
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;XPi is the 66-bit value of Pi from the Intel manual
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XPiHi equ 0C90FDAA2H
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XPiMid equ 02168C234H
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XPiLo equ 0C0000000H ;Extension of pi
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PiOver4exp equ -1 ;Pi/4 ~= 3/4, so exponent is -1
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TinyAngleExp equ -32 ;Smallest angle we bother with
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MaxAngleExp equ 63 ;Angle that's too big
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Trig1Result:
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;Trig function reduction routine used by functions returning 1 value
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;(FSIN and FCOS)
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;edi = [CURstk] = argument pointer
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;Argument has already been checked for zero.
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;ZF = (tag == bTAG_ZERO)
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jb TrigPrem
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;Tagged special
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mov al,EMSEG:[edi].bTAG
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cmp al,bTAG_DEN
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jz TrigDenorm
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add sp,4 ;Don't return to caller
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cmp al,bTAG_INF
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jnz SpclDestNotDen ;Check for Empty or NAN
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mov EMSEG:[SWcc],C2 ;Can't reduce infinity
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jmp ReturnIndefinite
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TrigDenorm:
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mov EMSEG:[CURerr],Denormal
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test EMSEG:[CWmask],Denormal ;Is denormal exception masked?
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jnz TrigPrem ;Yes, continue
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add sp,4 ;Don't return to caller
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TrigRet:
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ret
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Trig2Inf:
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mov EMSEG:[SWcc],C2 ;Can't reduce infinity
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jmp Trig2Indefinite
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Trig2StackOver:
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mov EMSEG:[SWcc],C1 ;Signal overflow
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Trig2StackUnder:
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mov EMSEG:[CURerr],Invalid+StackFlag
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Trig2Indefinite:
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add sp,4 ;Don't return to caller
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call ReturnIndefinite
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jz TrigRet ;Unmasked, don't change registers
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;Produce masked response
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mov EMSEG:[CURstk],esi ;Push stack
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mov edi,esi
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jmp ReturnIndefinite
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Trig2Special:
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cmp al,bTAG_DEN
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jz TrigDenorm
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cmp al,bTAG_INF
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jz Trig2Inf
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;Must be a NAN
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add sp,4 ;Don't return to caller
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call DestNAN
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jz TrigRet ;Unmasked, don't change registers
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;Produce masked response
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mov EMSEG:[CURstk],esi ;Push stack
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mov eax,EMSEG:[edi].ExpSgn
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mov EMSEG:[esi].ExpSgn,eax
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mov eax,EMSEG:[edi].lManHi
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mov EMSEG:[esi].lManHi,eax
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mov eax,EMSEG:[edi].lManLo
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mov EMSEG:[esi].lManLo,eax
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ret
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Trig2Zero:
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add sp,4 ;Don't return to caller
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mov EMSEG:[CURstk],esi
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mov edi,esi
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;Amazing coincidence: both FSINCOS and FPTAN return the same result for
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;a zero argument:
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; FSINCOS returns ST(0) = cos(0) = 1, ST(1) = sin(0) = 0.
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; FPTAN returns ST(0) = 1 always, ST(1) = tan(0) = 0.
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;Return zero has same sign as argument zero, so we don't need to touch
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;it -- just push +1.0.
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jmp ReturnOne
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TrigOutOfRange:
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mov EMSEG:[SWcc],C2 ;Signal argument not reduced
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add sp,4
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ret
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PrevStackWrap esi,Trig2 ;Tied to PrevStackElem below
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Trig2Result:
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;Trig function reduction routine used by functions returning 2 values
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;(FSINCOS and FPTAN)
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;edi = [CURstk] = argument pointer
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mov esi,edi
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PrevStackElem esi,Trig2 ;esi points to second result location
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mov al,EMSEG:[edi].bTAG ;Get tag
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cmp al,bTAG_EMPTY ;Stack underflow if empty
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jz Trig2StackUnder
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cmp EMSEG:[esi].bTAG,bTAG_EMPTY ;Stack overflow if not empty
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jnz Trig2StackOver
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cmp al,bTAG_ZERO ;Is it Special?
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ja Trig2Special
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jz Trig2Zero
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;Fall into TrigPrem
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;TrigPrem
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;This routine reduces an angle in radians to the range [0, pi/4].
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;Angles in odd-numbered octants have been subtracted from pi/4.
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;It uses a 66-bit value for pi, as required by the 387.
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;TrigPrem uses the same two-stage algorithm as FPREM (see
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;emfprem.asm). However, it is limited to an argument < 2^63.
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;Inputs:
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; edi = [CURstk]
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;Outputs:
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; ebx:esi = remainder, normalized
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; high ecx = exponent, cl = tag
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; al = octant
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; edi = [CURstk]
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TrigPrem:
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mov EMSEG:[Result],edi
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mov eax,EMSEG:[edi].lManLo
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mov edx,EMSEG:[edi].lManHi
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movsx ebx,EMSEG:[edi].wExp
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cmp ebx,MaxAngleExp
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jge TrigOutOfRange
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xor edi,edi ;Extend dividend
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xor esi,esi ;Quotient, in case we skip stage 1
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.erre PiOver4exp eq -1
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inc ebx ;Subtract exponent of pi/4
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jl ExitTrigPrem ;If dividend is smaller, return it.
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;We now know that 0 <= ExpDif < 64, so it fits in bl.
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cmp bl,31 ;Do we need to do stage 1?
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jl FitPi ;No, start stage 2
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;FPREM stage 1
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;Exponent difference is at least 31. Use 32-bit division to compute
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;quotient and exact remainder, reducing exponent difference by 31.
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;edx:eax = dividend
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;ebx = exponent difference
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;Shift dividend right one bit to be sure DIV instruction won't overflow
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;This means we'll be reducing the exponent difference by 31, not 32
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xor ebp,ebp ;Dividend extension
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shrd ebp,eax,1
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shrd eax,edx,1
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shr edx,1
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sub bl,31 ;Exponent reduced
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mov ecx,XPiHi
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div ecx ;Guess a quotient "digit"
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;Check out our guess.
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;Currently, remainder in edx = (high dividend) - (quotient * high pi).
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;(High dividend is the upper 64 bits--ebp has 1 bit.) The definition
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;of remainder is (all dividend) - (quotient * all pi). So if we
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;subtract (quotient * low pi) from edx:ebp, we'll get the true
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;remainder. If it's negative, our guess was too big.
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mov esi,eax ;Save quotient
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mov ecx,edx ;Save remainder
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;The pi/4 we use has two bits set below the first 64 bits. This means
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;we must add another 3/4 of the quotient into the amount to subtract,
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;which we'll compute by rounding the low 32 bits up 1, then subtracting
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;1/4 of quotient. But since we're computing the amount to subtract from
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;the remainder, we'll add the 1/4 of the quotient to the remainder instead
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;of subtracting it from the amount to subtract.
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.erre XPiLo eq (3 shl 30)
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mov eax,XPiMid+1
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mul esi ;Quotient * low pi
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;Note that ebp is either 0 or 800...00H
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shr ebp,30 ;Move down to low end
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shld ebp,esi,30 ;Move back up, adding 1/4 of quotient
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mov edi,esi ;Another copy of quotient
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shl edi,30 ;Keep last two bits
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;edx:eax has amount to subtract to get correct remainder from ecx:ebp:edi
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sub ebp,eax
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sbb ecx,edx ;Subtract from remainder
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mov eax,ebp
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mov edx,ecx ;Remainder back to edx:eax:edi
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jnc TrigPremNorm ;Was quotient OK?
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TrigCorrect:
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dec esi ;Quotient was too big
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add edi,XPiLo
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adc eax,XPiMid ;Add divisor back into remainder
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adc edx,XPiHi
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jnc TrigCorrect ;Repeat if quotient is still too big
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jmp TrigPremNorm
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;FPREM stage 2
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;Exponent difference is less than 32. Use restoring long division to
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;compute quotient bits until exponent difference is zero. Note that we
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;often get more than one bit/loop: BSR is used to scan off leading
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;zeros each time around. Since the divisor is normalized, we can
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;instantly compute a zero quotient bit for each leading zero bit.
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TrigPremLoop:
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;edx:eax:edi = dividend (remainder) minus pi/4
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;esi = quotient
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;ebx = exponent difference
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;If D is current dividend and p is pi/4, then we have edx:eax:edi = D - p,
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;which is negative. We want 2*D - p, which is positive.
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;2*D - p = 2*(D - p) + p.
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add edi,edi ;2*(D - p)
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adc eax,eax
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adc edx,edx
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add edi,XPiLo ;2*(D-p) + p = 2*D - p
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adc eax,XPiMid
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adc edx,XPiHi
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add esi,esi ;Double quotient too
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dec ebx ;Decrement exponent difference
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PiFit:
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inc esi
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TrigPremNorm:
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bsr ecx,edx ;Find first 1 bit
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jz TrigPremZero
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not cl
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and cl,1FH ;Convert bit no. to shift count
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sub ebx,ecx ;Reduce exponent difference
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jl TrigTooFar
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shld edx,eax,cl
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shld eax,edi,cl
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shl edi,cl ;Finish normalize shift
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shl esi,cl ;Shift quotient
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FitPi:
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;Dividend could be larger or smaller than divisor
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sub edi,XPiLo
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sbb eax,XPiMid
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sbb edx,XPiHi
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jnc PiFit
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;Couldn't subtract pi/2 from dividend.
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;edx:eax:edi = dividend - pi/4, which is negative
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or ebx,ebx ;Is exponent difference zero?
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jg TrigPremLoop
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;If quotient (octant number) is odd, we have subtracted an odd number of
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;pi/4's. However, simple angle reductions work in multiples of pi/2.
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;We will keep the extra pi/4 we just subtracted if the octant was odd.
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;This will give a result range of [-pi/4, pi/4].
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test esi,1 ;Is octant odd?
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jz EvenOctant
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NegPremResult:
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;-pi/4 < dividend < 0. Negate this since we use sign-magnitude representation.
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not edx ;96-bit negation
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not eax
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neg edi
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sbb eax,-1
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sbb edx,-1
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;May need to normalize
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bsr ecx,edx
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jz TrigNorm32
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lea ebx,[ebx+ecx-31] ;Fix up exponent for normalization
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not cl ;Convert bit no. to shift count
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TrigShortNorm:
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shld edx,eax,cl
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shld eax,edi,cl
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shl edi,cl ;Finish normalize shift
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RoundPrem:
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;Must round 66-bit result to 64 bits.
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;To perform "round even" when the round bit is set and the sticky bits
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;are zero, we treat the LSB as if it were a sticky bit. Thus if the LSB
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;is set, that will always force a round up (to even) if the round bit is
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;set. If the LSB is zero, then the sticky bits remain zero and we always
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;round down. This rounding rule is implemented by adding RoundBit-1
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;(7F..FFH), setting CY if round up.
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bt eax,0 ;Is mantissa even or odd? (set CY)
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adc edi,(1 shl 31)-1 ;Sum LSB & sticky bits--CY if round up
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adc eax,0
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adc edx,0
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ExitTrigPrem:
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;edx:eax = remainder, normalized
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;esi = quotient
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;ebx = exponent difference, zero or less
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.erre PiOver4exp eq -1
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dec ebx ;True exponent
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.erre bTAG_SNGL eq 0
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shrd ecx,ebx,16 ;Exponent to high ecx
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mov ebx,edx ;High mant. to ebx
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xchg esi,eax ;Low mant. to esi, octant to eax
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or esi,esi ;Any bits in low half?
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.erre bTAG_VALID eq 1
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.erre bTAG_SNGL eq 0
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setnz cl ;if low half==0 then cl=0 else cl=1
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mov edi,EMSEG:[CURstk]
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test EMSEG:[edi].bSgn,bSign ;Was angle negative?
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jnz FlipOct ;Yes, flip octant over
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ret
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FlipOct:
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;Angle was negative. Subtract octant from 7.
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neg al
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add al,7
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ret
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EvenOctant:
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;Restore dividend
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add edi,XPiLo
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adc eax,XPiMid
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adc edx,XPiHi
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jmp RoundPrem
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TrigTooFar:
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;Exponent difference in ebx went negative when reduced by shift count in ecx.
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;We need a quotient corresponding to exponent difference of zero.
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add ecx,ebx ;Compute previous exponent difference
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shl esi,cl ;Fix up quotient
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sub ecx,ebx ;Restore shift count
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test esi,1 ;Is octant odd?
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jz TrigShortNorm ;No, go normalize
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xor ebx,ebx ;Restore old exponent difference (zero)
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SubPiOver4:
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;We are here if exponent difference was zero and octant is odd.
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;As noted above, we need to reduce the angle by a multiple of pi/2,
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;not pi/4. We will subtract one more pi/4, which will make the
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;result range [-pi/4, pi/4].
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sub edi,XPiLo
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sbb eax,XPiMid
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sbb edx,XPiHi
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jmp NegPremResult
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TrigPremZero:
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;High dword of remainder is all zero, so we've reduced exponent difference
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;by 32 bits and overshot. We need a quotient corresponding to exponent
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;difference of zero, so we just shift it by the original difference. Then
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;we need to normalize the rest of the remainder.
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mov ecx,ebx ;Get exponent difference
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shl esi,cl ;Fix up quotient
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test esi,1 ;Is octant odd?
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jnz SubPiOver4 ;Yes, go subtract another pi/4
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TrigNorm32:
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bsr ecx,eax
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jz TinyTrig
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lea ebx,[ebx+ecx-31-32] ;Fix up exponent for normalization
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mov edx,eax
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mov eax,edi ;Shift left by 32 bits
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not cl ;Convert bit no. to shift count
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shld edx,eax,cl ;Normalize remainder
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shl eax,cl
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jmp ExitTrigPrem
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TinyTrig:
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;Upper 64 bits of remainder are all zero. We are assured that the extended
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;remainder is never zero, though.
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mov edx,edi ;Shift left 64 bits
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bsr ecx,edi
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lea ebx,[ebx+ecx-31-64] ;Fix up exponent for normalization
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not cl ;Convert bit no. to shift count
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shl edx,cl ;Normalize
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jmp ExitTrigPrem
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EM_ENTRY eFCOS
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eFCOS:
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and [esp].[OldLongStatus+4],NOT(C2 SHL 16) ;clear C2
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cmp EMSEG:[edi].bTAG,bTAG_ZERO
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jz ReturnOne
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call Trig1Result
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;ebx:esi,ecx = reduced argument
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;eax = octant
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mov ch,80H ;Assume negative
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test al,110B ;Negative in octants 2 - 5
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jpo @F ;Occurs when 1 of these bits are set
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xor ch,ch ;Actually positve
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@@:
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test al,011B ;Look for octants 0,3,4,7
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jpo TakeSine ;Use sine if not
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TakeCosine:
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cmp ecx,TinyAngleExp shl 16 ;Is angle really small?
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jl CosReturnOne ;cos(x) = 1 for tiny x
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CosNotTiny:
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mov edi,offset tCosPoly
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;Note that argument needs to be saved in ArgTemp (by EvalPolySetup) in case
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;we were called from eFSINCOS and we'll need the arg for the sine. Argument
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;is not needed for cosine, however (just its square).
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call EvalPolySetup ;In emftran.asm
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mov ch,EMSEG:[ArgTemp].bSgn ;Get sign we already figured out
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TransUnround:
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;The last operation performed a simple round nearest, without setting the
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;C1 status bit if round up occured. We reverse this last rounding now
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;so we can do the user's selected rounding mode. We also ensure that
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;the answer is never exact.
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sub eax,(1 shl 31)-1 ;Sum LSB & sticky bits--CY if round up
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jz UnroundExact ;Answer looks exact, but it's not
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sbb esi,0
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sbb ebx,0
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jns PolyDropExponent ;We had rounded up exponent too
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FinalTransRound:
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;A jump through [TransRound] is only valid if the number is known not to
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;underflow. Unmasked underflow requires [RoundMode] be set.
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mov edx,EMSEG:[TransRound]
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mov EMSEG:[RoundMode],edx
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call edx ;Perform user's rounding
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RestoreRound:
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;Restore rounding vectors
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mov EMSEG:[ZeroVector],offset SaveResult
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mov eax,EMSEG:[SavedRoundMode]
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mov EMSEG:[RoundMode],eax
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ret
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UnroundExact:
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inc eax ;Let's say our answer is a bit small
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jmp FinalTransRound
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PolyDropExponent:
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sub ecx,1 shl 16 ;Decrement exponent
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or ebx,1 shl 31 ;Set MSB
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jmp FinalTransRound
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SinRet:
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ret
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SaveTinySin:
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;Argument in ebx:esi,ecx is small enough so that sin(x) = x, which happens
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;when x - x^3/6 = x [or 1 - x^2/6 = 1]. Note that the infinitely precise
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;result is slightly less than the argument. To get the correct answer for
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;any rounding mode, we decrement the argument and set up for rounding.
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mov eax,-1 ;Set up rounding bits
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sub esi,1
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sbb ebx,0 ;Drop mantissa by one
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js FinalTransRound ;Still normalized?
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;mantissa must have been 800..000H, set it to 0FFF...FFFH and drop exponent
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mov ebx,eax ;ebx = -1
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sub ecx,1 shl 16 ;Drop exponent by one
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jmp FinalTransRound
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EM_ENTRY eFSIN
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eFSIN:
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and [esp].[OldLongStatus+4],NOT(C2 SHL 16) ;clear C2
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cmp EMSEG:[edi].bTAG,bTAG_ZERO
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jz SinRet ;Return zero for zero argument
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call Trig1Result
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mov ch,al
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shl ch,7-2 ;Move bit 2 to bit 7 as sign bit
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ReducedSine:
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;ebx:esi,ecx = reduced argument
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;ch = correct sign
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;eax = octant
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test al,011B ;Look for octants 0,3,4,7
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jpo TakeCosine ;Use cosine if not
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TakeSine:
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cmp ecx,TinyAngleExp shl 16 ;Is angle really small?
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jl SaveTinySin ;sin(x) = x for tiny x
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;The polynomial for sine is sin(x) = x * P(x^2). However, the degree zero
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;coefficient of P() is 1, so P() = R() + 1, where R() has no degree zero
|
|
;term. Thus sin(x) = x * [R(x^2) + 1] = x * R(x^2) + x.
|
|
|
|
;What's important here is that adding 1 to R(x^2) can blow away a lot of
|
|
;precision just before we do that last multiply by x. Note that x < pi/4 < 1,
|
|
;so that x^2 is often << 1. The precision is lost when R(x^2) is shifted
|
|
;right to align its binary point with 1.0. This can cause a loss of at
|
|
;least 1 bit of precision after the final multiply by x in addition to
|
|
;rounding errors.
|
|
|
|
;To avoid this precision loss, we use the alternate form given above,
|
|
;sin(x) = x * R(x^2) + x. Instead of adding 1.0 and multiplying by x,
|
|
;we multiply by x and add x--exactly the same level of difficulty. But
|
|
;the mulitply has all of R(x^2)'s precision available.
|
|
|
|
;Because the polynomial R() has no zero-degree term, we give EvalPoly
|
|
;one degree less (so we don't have to add zero as the last term).
|
|
;Then we have to multiply once more by x^2 since we left the loop early.
|
|
|
|
SineNotTiny:
|
|
mov edi,offset tSinPoly
|
|
call EvalPolySetup ;In emftran.asm
|
|
SineFinish:
|
|
|
|
ifdef NT386
|
|
mov edi,YFloatTemp
|
|
else
|
|
mov edi,offset edata:FloatTemp
|
|
endif
|
|
call PolyMulDouble ;Last coefficient in R(x^2)
|
|
|
|
ifdef NT386
|
|
mov edi,YArgTemp ;Point to original x
|
|
else
|
|
mov edi,offset edata:ArgTemp ;Point to original x
|
|
endif
|
|
|
|
call PolyMulDouble ;Compute x * R(x^2)
|
|
|
|
ifdef NT386
|
|
mov edi,YArgTemp ;Point to original x
|
|
else
|
|
mov edi,offset edata:ArgTemp ;Point to original x
|
|
endif
|
|
|
|
push offset TransUnround
|
|
jmp PolyAddDouble ;Compute x * R(x^2) + x
|
|
|
|
|
|
EM_ENTRY eFPTAN
|
|
eFPTAN:
|
|
and [esp].[OldLongStatus+4],NOT(C2 SHL 16) ;clear C2
|
|
call Trig2Result
|
|
push offset TanPushOne ; Push 1.0 when we're all done
|
|
;ebx:esi,ecx = reduced argument
|
|
;eax = octant
|
|
mov ch,al
|
|
shl ch,7-1 ;Move bit 1 to bit 7 as sign bit
|
|
;Note that ch bit 6 now has even/odd octant, which we'll need when we're
|
|
;done to see if we should take reciprocal.
|
|
cmp ecx,TinyAngleExp shl 16 ;Is angle really small?
|
|
jl TinyTan
|
|
mov edi,offset tTanPoly
|
|
call Eval2Poly ;In emftran.asm
|
|
mov edi,EMSEG:[CURstk] ;Point to first result
|
|
push offset TransUnround ;Return address of divide
|
|
test EMSEG:[ArgTemp].bSgn,0C0H ;Check low 2 bits of octant
|
|
;Given the reduced input range, the result can never overflow or underflow.
|
|
;It is must then be safe to assume neither operand is zero.
|
|
jpe DivDouble ;Tan() octants 0,3,4,7
|
|
jmp DivrDouble ;CoTan()
|
|
|
|
TinyTan:
|
|
test ch,0C0H ;Check low 2 bits of octant
|
|
jpe SaveTinySin ;Octants 0,3,4,7: tan(x) = x for tiny x
|
|
;Need reciprocal of reduced argument
|
|
mov edi,esi
|
|
mov esi,ebx ;Mantissa in esi:edi
|
|
mov ebx,ecx ;ExpSgn to ebx
|
|
mov edx,1 shl 31 ;Load 1.0
|
|
xor eax,eax
|
|
.erre TexpBias eq 0
|
|
xor ecx,ecx ;Sign and exponent are zero
|
|
;dividend mantissa in edx:eax, exponent in high ecx, sign in ch bit 7
|
|
;divisor mantissa in esi:edi, exponent in high ebx, sign in bh bit 7
|
|
push offset TransUnround ;Return address of divide
|
|
;Note that this can never overflow, because the reduced argument is never
|
|
;smaller than about 2^-65.
|
|
jmp DivDoubleReg
|
|
|
|
|
|
PrevStackWrap edi,Tan ;Tied to PrevStackElem below
|
|
|
|
TanPushOne:
|
|
PrevStackElem edi,Tan ;edi points to second result location
|
|
mov EMSEG:[CURstk],edi
|
|
ReturnOne:
|
|
mov EMSEG:[edi].lManLo,0
|
|
mov EMSEG:[edi].lManHi,1 shl 31
|
|
mov EMSEG:[edi].ExpSgn,(0-TexpBias) shl 16 + bTAG_SNGL
|
|
ret
|
|
|
|
|
|
PrevStackWrap edi,SinCos ;Tied to PrevStackElem below
|
|
|
|
eFSINCOS:
|
|
and [esp].[OldLongStatus+4],NOT(C2 SHL 16) ;clear C2
|
|
call Trig2Result
|
|
;Figure out signs
|
|
mov ch,al ;Start with sign of sine
|
|
shl ch,7-2 ;Move bit 2 to bit 7 as sign bit
|
|
mov ah,80H ;Assume sign of cosine is negative
|
|
test al,110B ;Negative in octants 2 - 5
|
|
jpo @F ;Occurs when 1 of these bits are set
|
|
xor ah,ah ;Actually positve
|
|
@@:
|
|
;ch = sign of sine
|
|
;ah = sign of cosine
|
|
cmp ecx,TinyAngleExp shl 16 ;Is angle really small?
|
|
jl TinySinCos
|
|
push eax ;Save octant and sign of cosine
|
|
call ReducedSine ;On exit, edi = [CURstk]
|
|
pop eax
|
|
;The Sin() funcion restored the rounding vectors to normal. Set them back.
|
|
mov EMSEG:[RoundMode],offset PolyRound
|
|
mov EMSEG:[ZeroVector],offset PolyZero
|
|
PrevStackElem edi,SinCos ;edi points to second result location
|
|
mov EMSEG:[CURstk],edi
|
|
mov EMSEG:[Result],edi
|
|
;Load x^2 back into registers
|
|
mov ecx,EMSEG:[FloatTemp].ExpSgn
|
|
mov ebx,EMSEG:[FloatTemp].lManHi
|
|
mov esi,EMSEG:[FloatTemp].lManLo
|
|
mov EMSEG:[ArgTemp].bSgn,ah ;Save sign
|
|
test al,011B ;Look for octants 0,3,4,7
|
|
jpo FastSine ;Use sine if not
|
|
mov edi,offset tCosPoly
|
|
call EvalPoly ;In emftran.asm
|
|
mov ch,EMSEG:[ArgTemp].bSgn ;Get sign we already figured out
|
|
jmp TransUnround
|
|
|
|
FastSine:
|
|
mov edi,offset tSinPoly
|
|
push offset SineFinish
|
|
jmp EvalPoly ;In emftran.asm
|
|
|
|
TinySinCos:
|
|
;ch = sign of sine
|
|
;ah = sign of cosine
|
|
;ebx:esi,high ecx = reduced argument
|
|
;edi = [CURstk]
|
|
test al,011B ;Look for octants 0,3,4,7
|
|
jpo TinyCosSin ;Take cosine first if not
|
|
push eax
|
|
call SaveTinySin ;For sine, arg is result
|
|
pop ecx
|
|
;edi = [CURstk]
|
|
;ch = sign of cosine
|
|
;Set cosine to 1.0
|
|
PrevStackElem edi,TinySinCos ;edi points to second result location
|
|
mov EMSEG:[CURstk],edi
|
|
mov EMSEG:[Result],edi
|
|
CosReturnOne:
|
|
;Cosine is nearly equal to 1.0. Put in next smaller value and round it.
|
|
mov ebx,-1
|
|
mov esi,ebx ;Set mantissa to -1
|
|
mov eax,ebx ;Set up rounding bits
|
|
.erre TexpBias eq 0
|
|
and ecx,bSign shl 8 ;Keep only sign
|
|
sub ecx,1 shl 16 ;Exponent of -1
|
|
;A jump through [TransRound] is only valid if the number is known not to
|
|
;underflow. Unmasked underflow requires [RoundMode] be set.
|
|
jmp EMSEG:[TransRound]
|
|
|
|
PrevStackWrap edi,TinySinCos
|
|
|
|
PrevStackWrap edi,TinyCosSin
|
|
|
|
TinyCosSin:
|
|
;Sine is nearly 1.0, cosine is argument
|
|
|
|
;ch = sign of sine
|
|
;ah = sign of cosine
|
|
;ebx:esi,high ecx = reduced argument
|
|
;edi = [CURstk]
|
|
xchg ah,ch ;Cosine sign to ch, sine sign to ah
|
|
push edi ;Save place for sine
|
|
PrevStackElem edi,TinyCosSin ;edi points to second result location
|
|
mov EMSEG:[CURstk],edi
|
|
mov EMSEG:[Result],edi
|
|
push eax
|
|
call SaveTinySin ;For sine, arg is result
|
|
pop ecx
|
|
;ch = sign of sine
|
|
pop EMSEG:[Result] ;Set up location for sine
|
|
jmp CosReturnOne
|
|
|
|
|
|
|
|
;********************* Polynomial Coefficients *********************
|
|
|
|
;These polynomial coefficients were all taken from "Computer Approximations"
|
|
;by J.F. Hart (reprinted 1978 w/corrections). All calculations and
|
|
;conversions to hexadecimal were done with a character-string calculator
|
|
;written in Visual Basic with precision set to 30 digits. Once the constants
|
|
;were typed into this file, all transfers were done with cut-and-paste
|
|
;operations to and from the calculator to help eliminate any typographical
|
|
;errors.
|
|
|
|
|
|
tCosPoly label word
|
|
|
|
;These constants are derived from Hart #3824: cos(x) = P(x^2),
|
|
;accurate to 19.45 digits over interval [0, pi/4]. The original
|
|
;constants in Hart required that the argument x be divided by pi/4.
|
|
;These constants have been scaled so this is no longer required.
|
|
;Scaling is done by multiplying the constant by a power of 4/pi.
|
|
;The power is given in the table.
|
|
|
|
dd 7 ;Degree seven
|
|
|
|
; Original Hart constant power Scaled constant
|
|
|
|
;-0.38577 62037 2 E-12 14 -0.113521232057839395845871741043E-10
|
|
;Hex value: 0.C7B56AF786699CF1BD13FD290 HFFDC
|
|
dq 0C7B56AF786699CF2H
|
|
dw (bSign shl 8)+bTAG_VALID,0FFDCH-1
|
|
|
|
;+0.11500 49702 4263 E-9 12 +0.208755551456778828747793797596E-8
|
|
;Hex value: 0.8F74AA3CCE49E68D6F5444A18 HFFE4
|
|
dq 08F74AA3CCE49E68DH
|
|
dw bTAG_VALID,0FFE4H-1
|
|
|
|
;-0.24611 36382 63700 5 E-7 10 -0.275573128656960822243472872247E-6
|
|
;Hex value: 0.93F27B7F10CC8A1703EFC8A04 HFFEB
|
|
dq 093F27B7F10CC8A17H
|
|
dw (bSign shl 8)+bTAG_VALID,0FFEBH-1
|
|
|
|
;+0.35908 60445 88581 953 E-5 8 +0.248015872828994630247806807317E-4
|
|
;Hex value: 0.D00D00CD6BB3ECD17E10D5830 HFFF1
|
|
dq 0D00D00CD6BB3ECD1H
|
|
dw bTAG_VALID,0FFF1H-1
|
|
|
|
;-0.32599 18869 26687 55044 E-3 6 -0.138888888888589604343951947246E-2
|
|
;Hex value: 0.B60B60B609B165894CFE522AC HFFF7
|
|
dq 0B60B60B609B16589H
|
|
dw (bSign shl 8)+bTAG_VALID,0FFF7H-1
|
|
|
|
;+0.15854 34424 38154 10897 54 E-1 4 +0.416666666666664302573692446873E-1
|
|
;Hex value: 0.AAAAAAAAAAA99A1AF53042B08 HFFFC
|
|
dq 0AAAAAAAAAAA99A1BH
|
|
dw bTAG_VALID,0FFFCH-1
|
|
|
|
;-0.30842 51375 34042 45242 414 E0 2 -0.499999999999999992843582920899E0
|
|
;Hex value: 0.FFFFFFFFFFFFFEF7F98D3BFA8 HFFFF
|
|
dq 0FFFFFFFFFFFFFEF8H
|
|
dw (bSign shl 8)+bTAG_VALID,0FFFFH-1
|
|
|
|
;+0.99999 99999 99999 99996 415 E0 0 (no change)
|
|
;Hex value 0.FFFFFFFFFFFFFFFF56B402618 H0
|
|
dq 0FFFFFFFFFFFFFFFFH
|
|
dw bTAG_VALID,00H-1
|
|
|
|
|
|
tSinPoly label word
|
|
|
|
;These constants are derived from Hart #3044: sin(x) = x * P(x^2),
|
|
;accurate to 20.73 digits over interval [0, pi/4]. The original
|
|
;constants in Hart required that the argument x be divided by pi/4.
|
|
;These constants have been scaled so this is no longer required.
|
|
;Scaling is done by multiplying the constant by a power of 4/pi.
|
|
;The power is given in the table.
|
|
|
|
dd 7-1 ;Degree seven, but the last coefficient
|
|
;is 1.0 and is not listed here.
|
|
|
|
; Original Hart constant power Scaled constant
|
|
|
|
;-0.20225 31292 93 E-13 15 -0.757786788401271156262125540409E-12
|
|
;Hex value: 0.D54C4AF2B524F0F2D6411C90A HFFD8
|
|
dq 0D54C4AF2B524F0F3H
|
|
dw (bSign shl 8)+bTAG_VALID,0FFD8H-1
|
|
|
|
;+0.69481 52035 0522 E-11 13 +0.160583476232246065559545749398E-9
|
|
;Hex value: 0.B0903AF085DA66030F16E43BC HFFE0
|
|
dq 0B0903AF085DA6603H
|
|
dw bTAG_VALID,0FFE0H-1
|
|
|
|
;-0.17572 47417 61708 06 E-8 11 -0.250521047382673309542092418731E-7
|
|
;Hex value: 0.D73229320D2AF05971AC96FF4 HFFE7
|
|
dq 0D73229320D2AF059H
|
|
dw (bSign shl 8)+bTAG_VALID,0FFE7H-1
|
|
|
|
;+0.31336 16889 17325 348 E-6 9 +0.275573192133901687156480447942E-5
|
|
;Hex value: 0.B8EF1D2984D2FBA28A9CC9DEE HFFEE
|
|
dq 0B8EF1D2984D2FBA3H
|
|
dw bTAG_VALID,0FFEEH-1
|
|
|
|
;-0.36576 20418 21464 00052 9 E-4 7 -0.198412698412531058609618529749E-3
|
|
;Hex value: 0.D00D00D00C3FDDD7916E5CB28 HFFF4
|
|
dq 0D00D00D00C3FDDD8H
|
|
dw (bSign shl 8)+bTAG_VALID,0FFF4H-1
|
|
|
|
;+0.24903 94570 19271 62752 519 E-2 5 +0.83333333333333203341753387264E-2
|
|
;Hex value: 0.8888888888884C95D619A0343 HFFFA
|
|
dq 08888888888884C96H
|
|
dw bTAG_VALID,0FFFAH-1
|
|
|
|
;-0.80745 51218 82807 81520 2582 E-1 3 -0.166666666666666666281276062229E0
|
|
;Hex value: 0.AAAAAAAAAAAAAA8E3AD80EAB8 HFFFE
|
|
dq 0AAAAAAAAAAAAAA8EH
|
|
dw (bSign shl 8)+bTAG_VALID,0FFFEH-1
|
|
|
|
;+0.78539 81633 97448 30961 41845 E0 1 +0.99999999999999999999812025812E0
|
|
;Hex value: 0.FFFFFFFFFFFFFFFFF71F88110 H0
|
|
; dq 8000000000000000H ;This constant of 1.0 omitted here.
|
|
; dw bTAG_VALID,0 ; It is handled in code.
|
|
|
|
|
|
tTanPoly label word
|
|
|
|
;These constants are derived from Hart #4286: tan(x) = x * P(x^2) / Q(x^2),
|
|
;accurate to 19.94 digits over interval [0, pi/4]. The original
|
|
;constants in Hart required that the argument x be divided by pi/4.
|
|
;These constants have been scaled so this is no longer required.
|
|
;Scaling is done by multiplying the constant by the same power of 4/pi
|
|
;as the power of x the constant is used on. However, the highest
|
|
;degree coefficient of Q() is 1, and after scaling this way it would
|
|
;become (4/pi)^8. In order to keep this coefficient equal to one,
|
|
;we scale everything again by (pi/4)^8. This scaling is partially
|
|
;canceled by the original scaling by powers of 4/pi, and the net
|
|
;resulting power of pi/4 is given in the table.
|
|
|
|
|
|
dd 3 ;First poly is degree 3
|
|
|
|
; Original Hart constant power Scaled constant
|
|
|
|
;-.45649 31943 86656 31873 96113 7 E2 1 -35.8528916474714232910463077546
|
|
;Hex value: 0.8F695C6D93AF6F97B6E022AB3 H6
|
|
dq 08F695C6D93AF6F98H
|
|
dw (bSign shl 8)+bTAG_VALID,06H-1
|
|
|
|
;+.14189 85425 27617 78388 00394 831 E5 3 +6874.60229709782436592720603503
|
|
;Hex value: 0.D6D4D181240D0D08C88DF4AA6 HD
|
|
dq 0D6D4D181240D0D09H
|
|
dw bTAG_VALID,0DH-1
|
|
|
|
;-.89588 84400 67680 41087 29639 541 E6 5 -267733.884797157298951145495276
|
|
;Hex value: 0.82BABC504220C62B1D0722684 H13
|
|
dq 082BABC504220C62BH
|
|
dw (bSign shl 8)+bTAG_VALID,013H-1
|
|
|
|
;+.10888 60043 72816 87521 38857 983 E8 7 +2007248.9111748838841548144685
|
|
;Hex value: 0.F506874A160EB9C0994AADD6A H15
|
|
dq 0F506874A160EB9C1H
|
|
dw bTAG_VALID,015H-1
|
|
|
|
|
|
|
|
dd 4 ;Second poly is degree 4
|
|
;NOTE: Eval2Poly assumes the first coefficient is 1.0, so it is omitted
|
|
|
|
; Original Hart constant power Scaled constant
|
|
|
|
;-.10146 56190 25288 53387 54401 947 E4 2 -625.890950057027419879480354834
|
|
;Hex value: 0.9C790553635355A95241A5324 HA
|
|
dq 09C790553635355A9H
|
|
dw (bSign shl 8)+bTAG_VALID,0AH-1
|
|
|
|
;+.13538 27128 05119 09382 89294 872 E6 4 +51513.6992033752080924797647367
|
|
;Hex value: 0.C939B2FEFE0DC585E649870FE H10
|
|
dq 0C939B2FEFE0DC586H
|
|
dw bTAG_VALID,010H-1
|
|
|
|
;-.39913 09518 03516 51504 43427 94 E7 6 -936816.855188785264866481436899
|
|
;Hex value: 0.E4B70DAEDA6F89E5A7CE626FA H14
|
|
dq 0E4B70DAEDA6F89E6H
|
|
dw (bSign shl 8)+bTAG_VALID,014H-1
|
|
|
|
;+.13863 79666 35676 29165 33913 361 E8 8 +2007248.91117488388417770850458
|
|
;Hex value: 0.F506874A160EB9C0CCD8313BC H15
|
|
dq 0F506874A160EB9C1H
|
|
dw bTAG_VALID,015H-1
|