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|| setox.sa 3.1 12/10/90|| The entry point setox computes the exponential of a value.| setoxd does the same except the input value is a denormalized| number. setoxm1 computes exp(X)-1, and setoxm1d computes| exp(X)-1 for denormalized X.|| INPUT| -----| Double-extended value in memory location pointed to by address| register a0.|| OUTPUT| ------| exp(X) or exp(X)-1 returned in floating-point register fp0.|| ACCURACY and MONOTONICITY| -------------------------| The returned result is within 0.85 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.|| SPEED| -----| Two timings are measured, both in the copy-back mode. The| first one is measured when the function is invoked the first time| (so the instructions and data are not in cache), and the| second one is measured when the function is reinvoked at the same| input argument.|| The program setox takes approximately 210/190 cycles for input| argument X whose magnitude is less than 16380 log2, which| is the usual situation. For the less common arguments,| depending on their values, the program may run faster or slower --| but no worse than 10% slower even in the extreme cases.|| The program setoxm1 takes approximately ???/??? cycles for input| argument X, 0.25 <= |X| < 70log2. For |X| < 0.25, it takes| approximately ???/??? cycles. For the less common arguments,| depending on their values, the program may run faster or slower --| but no worse than 10% slower even in the extreme cases.|| ALGORITHM and IMPLEMENTATION NOTES| ----------------------------------|| setoxd| ------| Step 1. Set ans := 1.0|| Step 2. Return ans := ans + sign(X)*2^(-126). Exit.| Notes: This will always generate one exception -- inexact.||| setox| -----|| Step 1. Filter out extreme cases of input argument.| 1.1 If |X| >= 2^(-65), go to Step 1.3.| 1.2 Go to Step 7.| 1.3 If |X| < 16380 log(2), go to Step 2.| 1.4 Go to Step 8.| Notes: The usual case should take the branches 1.1 -> 1.3 -> 2.| To avoid the use of floating-point comparisons, a| compact representation of |X| is used. This format is a| 32-bit integer, the upper (more significant) 16 bits are| the sign and biased exponent field of |X|; the lower 16| bits are the 16 most significant fraction (including the| explicit bit) bits of |X|. Consequently, the comparisons| in Steps 1.1 and 1.3 can be performed by integer comparison.| Note also that the constant 16380 log(2) used in Step 1.3| is also in the compact form. Thus taking the branch| to Step 2 guarantees |X| < 16380 log(2). There is no harm| to have a small number of cases where |X| is less than,| but close to, 16380 log(2) and the branch to Step 9 is| taken.|| Step 2. Calculate N = round-to-nearest-int( X * 64/log2 ).| 2.1 Set AdjFlag := 0 (indicates the branch 1.3 -> 2 was taken)| 2.2 N := round-to-nearest-integer( X * 64/log2 ).| 2.3 Calculate J = N mod 64; so J = 0,1,2,..., or 63.| 2.4 Calculate M = (N - J)/64; so N = 64M + J.| 2.5 Calculate the address of the stored value of 2^(J/64).| 2.6 Create the value Scale = 2^M.| Notes: The calculation in 2.2 is really performed by|| Z := X * constant| N := round-to-nearest-integer(Z)|| where|| constant := single-precision( 64/log 2 ).|| Using a single-precision constant avoids memory access.| Another effect of using a single-precision "constant" is| that the calculated value Z is|| Z = X*(64/log2)*(1+eps), |eps| <= 2^(-24).|| This error has to be considered later in Steps 3 and 4.|| Step 3. Calculate X - N*log2/64.| 3.1 R := X + N*L1, where L1 := single-precision(-log2/64).| 3.2 R := R + N*L2, L2 := extended-precision(-log2/64 - L1).| Notes: a) The way L1 and L2 are chosen ensures L1+L2 approximate| the value -log2/64 to 88 bits of accuracy.| b) N*L1 is exact because N is no longer than 22 bits and| L1 is no longer than 24 bits.| c) The calculation X+N*L1 is also exact due to cancellation.| Thus, R is practically X+N(L1+L2) to full 64 bits.| d) It is important to estimate how large can |R| be after| Step 3.2.|| N = rnd-to-int( X*64/log2 (1+eps) ), |eps|<=2^(-24)| X*64/log2 (1+eps) = N + f, |f| <= 0.5| X*64/log2 - N = f - eps*X 64/log2| X - N*log2/64 = f*log2/64 - eps*X||| Now |X| <= 16446 log2, thus|| |X - N*log2/64| <= (0.5 + 16446/2^(18))*log2/64| <= 0.57 log2/64.| This bound will be used in Step 4.|| Step 4. Approximate exp(R)-1 by a polynomial| p = R + R*R*(A1 + R*(A2 + R*(A3 + R*(A4 + R*A5))))| Notes: a) In order to reduce memory access, the coefficients are| made as "short" as possible: A1 (which is 1/2), A4 and A5| are single precision; A2 and A3 are double precision.| b) Even with the restrictions above,| |p - (exp(R)-1)| < 2^(-68.8) for all |R| <= 0.0062.| Note that 0.0062 is slightly bigger than 0.57 log2/64.| c) To fully utilize the pipeline, p is separated into| two independent pieces of roughly equal complexities| p = [ R + R*S*(A2 + S*A4) ] +| [ S*(A1 + S*(A3 + S*A5)) ]| where S = R*R.|| Step 5. Compute 2^(J/64)*exp(R) = 2^(J/64)*(1+p) by| ans := T + ( T*p + t)| where T and t are the stored values for 2^(J/64).| Notes: 2^(J/64) is stored as T and t where T+t approximates| 2^(J/64) to roughly 85 bits; T is in extended precision| and t is in single precision. Note also that T is rounded| to 62 bits so that the last two bits of T are zero. The| reason for such a special form is that T-1, T-2, and T-8| will all be exact --- a property that will give much| more accurate computation of the function EXPM1.|| Step 6. Reconstruction of exp(X)| exp(X) = 2^M * 2^(J/64) * exp(R).| 6.1 If AdjFlag = 0, go to 6.3| 6.2 ans := ans * AdjScale| 6.3 Restore the user FPCR| 6.4 Return ans := ans * Scale. Exit.| Notes: If AdjFlag = 0, we have X = Mlog2 + Jlog2/64 + R,| |M| <= 16380, and Scale = 2^M. Moreover, exp(X) will| neither overflow nor underflow. If AdjFlag = 1, that| means that| X = (M1+M)log2 + Jlog2/64 + R, |M1+M| >= 16380.| Hence, exp(X) may overflow or underflow or neither.| When that is the case, AdjScale = 2^(M1) where M1 is| approximately M. Thus 6.2 will never cause over/underflow.| Possible exception in 6.4 is overflow or underflow.| The inexact exception is not generated in 6.4. Although| one can argue that the inexact flag should always be| raised, to simulate that exception cost to much than the| flag is worth in practical uses.|| Step 7. Return 1 + X.| 7.1 ans := X| 7.2 Restore user FPCR.| 7.3 Return ans := 1 + ans. Exit| Notes: For non-zero X, the inexact exception will always be| raised by 7.3. That is the only exception raised by 7.3.| Note also that we use the FMOVEM instruction to move X| in Step 7.1 to avoid unnecessary trapping. (Although| the FMOVEM may not seem relevant since X is normalized,| the precaution will be useful in the library version of| this code where the separate entry for denormalized inputs| will be done away with.)|| Step 8. Handle exp(X) where |X| >= 16380log2.| 8.1 If |X| > 16480 log2, go to Step 9.| (mimic 2.2 - 2.6)| 8.2 N := round-to-integer( X * 64/log2 )| 8.3 Calculate J = N mod 64, J = 0,1,...,63| 8.4 K := (N-J)/64, M1 := truncate(K/2), M = K-M1, AdjFlag := 1.| 8.5 Calculate the address of the stored value 2^(J/64).| 8.6 Create the values Scale = 2^M, AdjScale = 2^M1.| 8.7 Go to Step 3.| Notes: Refer to notes for 2.2 - 2.6.|| Step 9. Handle exp(X), |X| > 16480 log2.| 9.1 If X < 0, go to 9.3| 9.2 ans := Huge, go to 9.4| 9.3 ans := Tiny.| 9.4 Restore user FPCR.| 9.5 Return ans := ans * ans. Exit.| Notes: Exp(X) will surely overflow or underflow, depending on| X's sign. "Huge" and "Tiny" are respectively large/tiny| extended-precision numbers whose square over/underflow| with an inexact result. Thus, 9.5 always raises the| inexact together with either overflow or underflow.||| setoxm1d| --------|| Step 1. Set ans := 0|| Step 2. Return ans := X + ans. Exit.| Notes: This will return X with the appropriate rounding| precision prescribed by the user FPCR.|| setoxm1| -------|| Step 1. Check |X|| 1.1 If |X| >= 1/4, go to Step 1.3.| 1.2 Go to Step 7.| 1.3 If |X| < 70 log(2), go to Step 2.| 1.4 Go to Step 10.| Notes: The usual case should take the branches 1.1 -> 1.3 -> 2.| However, it is conceivable |X| can be small very often| because EXPM1 is intended to evaluate exp(X)-1 accurately| when |X| is small. For further details on the comparisons,| see the notes on Step 1 of setox.|| Step 2. Calculate N = round-to-nearest-int( X * 64/log2 ).| 2.1 N := round-to-nearest-integer( X * 64/log2 ).| 2.2 Calculate J = N mod 64; so J = 0,1,2,..., or 63.| 2.3 Calculate M = (N - J)/64; so N = 64M + J.| 2.4 Calculate the address of the stored value of 2^(J/64).| 2.5 Create the values Sc = 2^M and OnebySc := -2^(-M).| Notes: See the notes on Step 2 of setox.|| Step 3. Calculate X - N*log2/64.| 3.1 R := X + N*L1, where L1 := single-precision(-log2/64).| 3.2 R := R + N*L2, L2 := extended-precision(-log2/64 - L1).| Notes: Applying the analysis of Step 3 of setox in this case| shows that |R| <= 0.0055 (note that |X| <= 70 log2 in| this case).|| Step 4. Approximate exp(R)-1 by a polynomial| p = R+R*R*(A1+R*(A2+R*(A3+R*(A4+R*(A5+R*A6)))))| Notes: a) In order to reduce memory access, the coefficients are| made as "short" as possible: A1 (which is 1/2), A5 and A6| are single precision; A2, A3 and A4 are double precision.| b) Even with the restriction above,| |p - (exp(R)-1)| < |R| * 2^(-72.7)| for all |R| <= 0.0055.| c) To fully utilize the pipeline, p is separated into| two independent pieces of roughly equal complexity| p = [ R*S*(A2 + S*(A4 + S*A6)) ] +| [ R + S*(A1 + S*(A3 + S*A5)) ]| where S = R*R.|| Step 5. Compute 2^(J/64)*p by| p := T*p| where T and t are the stored values for 2^(J/64).| Notes: 2^(J/64) is stored as T and t where T+t approximates| 2^(J/64) to roughly 85 bits; T is in extended precision| and t is in single precision. Note also that T is rounded| to 62 bits so that the last two bits of T are zero. The| reason for such a special form is that T-1, T-2, and T-8| will all be exact --- a property that will be exploited| in Step 6 below. The total relative error in p is no| bigger than 2^(-67.7) compared to the final result.|| Step 6. Reconstruction of exp(X)-1| exp(X)-1 = 2^M * ( 2^(J/64) + p - 2^(-M) ).| 6.1 If M <= 63, go to Step 6.3.| 6.2 ans := T + (p + (t + OnebySc)). Go to 6.6| 6.3 If M >= -3, go to 6.5.| 6.4 ans := (T + (p + t)) + OnebySc. Go to 6.6| 6.5 ans := (T + OnebySc) + (p + t).| 6.6 Restore user FPCR.| 6.7 Return ans := Sc * ans. Exit.| Notes: The various arrangements of the expressions give accurate| evaluations.|| Step 7. exp(X)-1 for |X| < 1/4.| 7.1 If |X| >= 2^(-65), go to Step 9.| 7.2 Go to Step 8.|| Step 8. Calculate exp(X)-1, |X| < 2^(-65).| 8.1 If |X| < 2^(-16312), goto 8.3| 8.2 Restore FPCR; return ans := X - 2^(-16382). Exit.| 8.3 X := X * 2^(140).| 8.4 Restore FPCR; ans := ans - 2^(-16382).| Return ans := ans*2^(140). Exit| Notes: The idea is to return "X - tiny" under the user| precision and rounding modes. To avoid unnecessary| inefficiency, we stay away from denormalized numbers the| best we can. For |X| >= 2^(-16312), the straightforward| 8.2 generates the inexact exception as the case warrants.|| Step 9. Calculate exp(X)-1, |X| < 1/4, by a polynomial| p = X + X*X*(B1 + X*(B2 + ... + X*B12))| Notes: a) In order to reduce memory access, the coefficients are| made as "short" as possible: B1 (which is 1/2), B9 to B12| are single precision; B3 to B8 are double precision; and| B2 is double extended.| b) Even with the restriction above,| |p - (exp(X)-1)| < |X| 2^(-70.6)| for all |X| <= 0.251.| Note that 0.251 is slightly bigger than 1/4.| c) To fully preserve accuracy, the polynomial is computed| as X + ( S*B1 + Q ) where S = X*X and| Q = X*S*(B2 + X*(B3 + ... + X*B12))| d) To fully utilize the pipeline, Q is separated into| two independent pieces of roughly equal complexity| Q = [ X*S*(B2 + S*(B4 + ... + S*B12)) ] +| [ S*S*(B3 + S*(B5 + ... + S*B11)) ]|| Step 10. Calculate exp(X)-1 for |X| >= 70 log 2.| 10.1 If X >= 70log2 , exp(X) - 1 = exp(X) for all practical| purposes. Therefore, go to Step 1 of setox.| 10.2 If X <= -70log2, exp(X) - 1 = -1 for all practical purposes.| ans := -1| Restore user FPCR| Return ans := ans + 2^(-126). Exit.| Notes: 10.2 will always create an inexact and return -1 + tiny| in the user rounding precision and mode.||| Copyright (C) Motorola, Inc. 1990| All Rights Reserved|| THIS IS UNPUBLISHED PROPRIETARY SOURCE CODE OF MOTOROLA| The copyright notice above does not evidence any| actual or intended publication of such source code.|setox idnt 2,1 | Motorola 040 Floating Point Software Package|section 8.include "fpsp.h"L2: .long 0x3FDC0000,0x82E30865,0x4361C4C6,0x00000000EXPA3: .long 0x3FA55555,0x55554431EXPA2: .long 0x3FC55555,0x55554018HUGE: .long 0x7FFE0000,0xFFFFFFFF,0xFFFFFFFF,0x00000000TINY: .long 0x00010000,0xFFFFFFFF,0xFFFFFFFF,0x00000000EM1A4: .long 0x3F811111,0x11174385EM1A3: .long 0x3FA55555,0x55554F5AEM1A2: .long 0x3FC55555,0x55555555,0x00000000,0x00000000EM1B8: .long 0x3EC71DE3,0xA5774682EM1B7: .long 0x3EFA01A0,0x19D7CB68EM1B6: .long 0x3F2A01A0,0x1A019DF3EM1B5: .long 0x3F56C16C,0x16C170E2EM1B4: .long 0x3F811111,0x11111111EM1B3: .long 0x3FA55555,0x55555555EM1B2: .long 0x3FFC0000,0xAAAAAAAA,0xAAAAAAAB.long 0x00000000TWO140: .long 0x48B00000,0x00000000TWON140: .long 0x37300000,0x00000000EXPTBL:.long 0x3FFF0000,0x80000000,0x00000000,0x00000000.long 0x3FFF0000,0x8164D1F3,0xBC030774,0x9F841A9B.long 0x3FFF0000,0x82CD8698,0xAC2BA1D8,0x9FC1D5B9.long 0x3FFF0000,0x843A28C3,0xACDE4048,0xA0728369.long 0x3FFF0000,0x85AAC367,0xCC487B14,0x1FC5C95C.long 0x3FFF0000,0x871F6196,0x9E8D1010,0x1EE85C9F.long 0x3FFF0000,0x88980E80,0x92DA8528,0x9FA20729.long 0x3FFF0000,0x8A14D575,0x496EFD9C,0xA07BF9AF.long 0x3FFF0000,0x8B95C1E3,0xEA8BD6E8,0xA0020DCF.long 0x3FFF0000,0x8D1ADF5B,0x7E5BA9E4,0x205A63DA.long 0x3FFF0000,0x8EA4398B,0x45CD53C0,0x1EB70051.long 0x3FFF0000,0x9031DC43,0x1466B1DC,0x1F6EB029.long 0x3FFF0000,0x91C3D373,0xAB11C338,0xA0781494.long 0x3FFF0000,0x935A2B2F,0x13E6E92C,0x9EB319B0.long 0x3FFF0000,0x94F4EFA8,0xFEF70960,0x2017457D.long 0x3FFF0000,0x96942D37,0x20185A00,0x1F11D537.long 0x3FFF0000,0x9837F051,0x8DB8A970,0x9FB952DD.long 0x3FFF0000,0x99E04593,0x20B7FA64,0x1FE43087.long 0x3FFF0000,0x9B8D39B9,0xD54E5538,0x1FA2A818.long 0x3FFF0000,0x9D3ED9A7,0x2CFFB750,0x1FDE494D.long 0x3FFF0000,0x9EF53260,0x91A111AC,0x20504890.long 0x3FFF0000,0xA0B0510F,0xB9714FC4,0xA073691C.long 0x3FFF0000,0xA2704303,0x0C496818,0x1F9B7A05.long 0x3FFF0000,0xA43515AE,0x09E680A0,0xA0797126.long 0x3FFF0000,0xA5FED6A9,0xB15138EC,0xA071A140.long 0x3FFF0000,0xA7CD93B4,0xE9653568,0x204F62DA.long 0x3FFF0000,0xA9A15AB4,0xEA7C0EF8,0x1F283C4A.long 0x3FFF0000,0xAB7A39B5,0xA93ED338,0x9F9A7FDC.long 0x3FFF0000,0xAD583EEA,0x42A14AC8,0xA05B3FAC.long 0x3FFF0000,0xAF3B78AD,0x690A4374,0x1FDF2610.long 0x3FFF0000,0xB123F581,0xD2AC2590,0x9F705F90.long 0x3FFF0000,0xB311C412,0xA9112488,0x201F678A.long 0x3FFF0000,0xB504F333,0xF9DE6484,0x1F32FB13.long 0x3FFF0000,0xB6FD91E3,0x28D17790,0x20038B30.long 0x3FFF0000,0xB8FBAF47,0x62FB9EE8,0x200DC3CC.long 0x3FFF0000,0xBAFF5AB2,0x133E45FC,0x9F8B2AE6.long 0x3FFF0000,0xBD08A39F,0x580C36C0,0xA02BBF70.long 0x3FFF0000,0xBF1799B6,0x7A731084,0xA00BF518.long 0x3FFF0000,0xC12C4CCA,0x66709458,0xA041DD41.long 0x3FFF0000,0xC346CCDA,0x24976408,0x9FDF137B.long 0x3FFF0000,0xC5672A11,0x5506DADC,0x201F1568.long 0x3FFF0000,0xC78D74C8,0xABB9B15C,0x1FC13A2E.long 0x3FFF0000,0xC9B9BD86,0x6E2F27A4,0xA03F8F03.long 0x3FFF0000,0xCBEC14FE,0xF2727C5C,0x1FF4907D.long 0x3FFF0000,0xCE248C15,0x1F8480E4,0x9E6E53E4.long 0x3FFF0000,0xD06333DA,0xEF2B2594,0x1FD6D45C.long 0x3FFF0000,0xD2A81D91,0xF12AE45C,0xA076EDB9.long 0x3FFF0000,0xD4F35AAB,0xCFEDFA20,0x9FA6DE21.long 0x3FFF0000,0xD744FCCA,0xD69D6AF4,0x1EE69A2F.long 0x3FFF0000,0xD99D15C2,0x78AFD7B4,0x207F439F.long 0x3FFF0000,0xDBFBB797,0xDAF23754,0x201EC207.long 0x3FFF0000,0xDE60F482,0x5E0E9124,0x9E8BE175.long 0x3FFF0000,0xE0CCDEEC,0x2A94E110,0x20032C4B.long 0x3FFF0000,0xE33F8972,0xBE8A5A50,0x2004DFF5.long 0x3FFF0000,0xE5B906E7,0x7C8348A8,0x1E72F47A.long 0x3FFF0000,0xE8396A50,0x3C4BDC68,0x1F722F22.long 0x3FFF0000,0xEAC0C6E7,0xDD243930,0xA017E945.long 0x3FFF0000,0xED4F301E,0xD9942B84,0x1F401A5B.long 0x3FFF0000,0xEFE4B99B,0xDCDAF5CC,0x9FB9A9E3.long 0x3FFF0000,0xF281773C,0x59FFB138,0x20744C05.long 0x3FFF0000,0xF5257D15,0x2486CC2C,0x1F773A19.long 0x3FFF0000,0xF7D0DF73,0x0AD13BB8,0x1FFE90D5.long 0x3FFF0000,0xFA83B2DB,0x722A033C,0xA041ED22.long 0x3FFF0000,0xFD3E0C0C,0xF486C174,0x1F853F3A.set ADJFLAG,L_SCR2.set SCALE,FP_SCR1.set ADJSCALE,FP_SCR2.set SC,FP_SCR3.set ONEBYSC,FP_SCR4| xref t_frcinx|xref t_extdnrm|xref t_unfl|xref t_ovfl.global setoxdsetoxd:|--entry point for EXP(X), X is denormalizedmovel (%a0),%d0andil #0x80000000,%d0oril #0x00800000,%d0 | ...sign(X)*2^(-126)movel %d0,-(%sp)fmoves #0x3F800000,%fp0fmovel %d1,%fpcrfadds (%sp)+,%fp0bra t_frcinx.global setoxsetox:|--entry point for EXP(X), here X is finite, non-zero, and not NaN's|--Step 1.movel (%a0),%d0 | ...load part of input Xandil #0x7FFF0000,%d0 | ...biased expo. of Xcmpil #0x3FBE0000,%d0 | ...2^(-65)bges EXPC1 | ...normal casebra EXPSMEXPC1:|--The case |X| >= 2^(-65)movew 4(%a0),%d0 | ...expo. and partial sig. of |X|cmpil #0x400CB167,%d0 | ...16380 log2 trunc. 16 bitsblts EXPMAIN | ...normal casebra EXPBIGEXPMAIN:|--Step 2.|--This is the normal branch: 2^(-65) <= |X| < 16380 log2.fmovex (%a0),%fp0 | ...load input from (a0)fmovex %fp0,%fp1fmuls #0x42B8AA3B,%fp0 | ...64/log2 * Xfmovemx %fp2-%fp2/%fp3,-(%a7) | ...save fp2movel #0,ADJFLAG(%a6)fmovel %fp0,%d0 | ...N = int( X * 64/log2 )lea EXPTBL,%a1fmovel %d0,%fp0 | ...convert to floating-formatmovel %d0,L_SCR1(%a6) | ...save N temporarilyandil #0x3F,%d0 | ...D0 is J = N mod 64lsll #4,%d0addal %d0,%a1 | ...address of 2^(J/64)movel L_SCR1(%a6),%d0asrl #6,%d0 | ...D0 is Maddiw #0x3FFF,%d0 | ...biased expo. of 2^(M)movew L2,L_SCR1(%a6) | ...prefetch L2, no need in CBEXPCONT1:|--Step 3.|--fp1,fp2 saved on the stack. fp0 is N, fp1 is X,|--a0 points to 2^(J/64), D0 is biased expo. of 2^(M)fmovex %fp0,%fp2fmuls #0xBC317218,%fp0 | ...N * L1, L1 = lead(-log2/64)fmulx L2,%fp2 | ...N * L2, L1+L2 = -log2/64faddx %fp1,%fp0 | ...X + N*L1faddx %fp2,%fp0 | ...fp0 is R, reduced arg.| MOVE.W #$3FA5,EXPA3 ...load EXPA3 in cache|--Step 4.|--WE NOW COMPUTE EXP(R)-1 BY A POLYNOMIAL|-- R + R*R*(A1 + R*(A2 + R*(A3 + R*(A4 + R*A5))))|--TO FULLY UTILIZE THE PIPELINE, WE COMPUTE S = R*R|--[R+R*S*(A2+S*A4)] + [S*(A1+S*(A3+S*A5))]fmovex %fp0,%fp1fmulx %fp1,%fp1 | ...fp1 IS S = R*Rfmoves #0x3AB60B70,%fp2 | ...fp2 IS A5| MOVE.W #0,2(%a1) ...load 2^(J/64) in cachefmulx %fp1,%fp2 | ...fp2 IS S*A5fmovex %fp1,%fp3fmuls #0x3C088895,%fp3 | ...fp3 IS S*A4faddd EXPA3,%fp2 | ...fp2 IS A3+S*A5faddd EXPA2,%fp3 | ...fp3 IS A2+S*A4fmulx %fp1,%fp2 | ...fp2 IS S*(A3+S*A5)movew %d0,SCALE(%a6) | ...SCALE is 2^(M) in extendedclrw SCALE+2(%a6)movel #0x80000000,SCALE+4(%a6)clrl SCALE+8(%a6)fmulx %fp1,%fp3 | ...fp3 IS S*(A2+S*A4)fadds #0x3F000000,%fp2 | ...fp2 IS A1+S*(A3+S*A5)fmulx %fp0,%fp3 | ...fp3 IS R*S*(A2+S*A4)fmulx %fp1,%fp2 | ...fp2 IS S*(A1+S*(A3+S*A5))faddx %fp3,%fp0 | ...fp0 IS R+R*S*(A2+S*A4),| ...fp3 releasedfmovex (%a1)+,%fp1 | ...fp1 is lead. pt. of 2^(J/64)faddx %fp2,%fp0 | ...fp0 is EXP(R) - 1| ...fp2 released|--Step 5|--final reconstruction process|--EXP(X) = 2^M * ( 2^(J/64) + 2^(J/64)*(EXP(R)-1) )fmulx %fp1,%fp0 | ...2^(J/64)*(Exp(R)-1)fmovemx (%a7)+,%fp2-%fp2/%fp3 | ...fp2 restoredfadds (%a1),%fp0 | ...accurate 2^(J/64)faddx %fp1,%fp0 | ...2^(J/64) + 2^(J/64)*...movel ADJFLAG(%a6),%d0|--Step 6tstl %d0beqs NORMALADJUST:fmulx ADJSCALE(%a6),%fp0NORMAL:fmovel %d1,%FPCR | ...restore user FPCRfmulx SCALE(%a6),%fp0 | ...multiply 2^(M)bra t_frcinxEXPSM:|--Step 7fmovemx (%a0),%fp0-%fp0 | ...in case X is denormalizedfmovel %d1,%FPCRfadds #0x3F800000,%fp0 | ...1+X in user modebra t_frcinxEXPBIG:|--Step 8cmpil #0x400CB27C,%d0 | ...16480 log2bgts EXP2BIG|--Steps 8.2 -- 8.6fmovex (%a0),%fp0 | ...load input from (a0)fmovex %fp0,%fp1fmuls #0x42B8AA3B,%fp0 | ...64/log2 * Xfmovemx %fp2-%fp2/%fp3,-(%a7) | ...save fp2movel #1,ADJFLAG(%a6)fmovel %fp0,%d0 | ...N = int( X * 64/log2 )lea EXPTBL,%a1fmovel %d0,%fp0 | ...convert to floating-formatmovel %d0,L_SCR1(%a6) | ...save N temporarilyandil #0x3F,%d0 | ...D0 is J = N mod 64lsll #4,%d0addal %d0,%a1 | ...address of 2^(J/64)movel L_SCR1(%a6),%d0asrl #6,%d0 | ...D0 is Kmovel %d0,L_SCR1(%a6) | ...save K temporarilyasrl #1,%d0 | ...D0 is M1subl %d0,L_SCR1(%a6) | ...a1 is Maddiw #0x3FFF,%d0 | ...biased expo. of 2^(M1)movew %d0,ADJSCALE(%a6) | ...ADJSCALE := 2^(M1)clrw ADJSCALE+2(%a6)movel #0x80000000,ADJSCALE+4(%a6)clrl ADJSCALE+8(%a6)movel L_SCR1(%a6),%d0 | ...D0 is Maddiw #0x3FFF,%d0 | ...biased expo. of 2^(M)bra EXPCONT1 | ...go back to Step 3EXP2BIG:|--Step 9fmovel %d1,%FPCRmovel (%a0),%d0bclrb #sign_bit,(%a0) | ...setox always returns positivecmpil #0,%d0blt t_unflbra t_ovfl.global setoxm1dsetoxm1d:|--entry point for EXPM1(X), here X is denormalized|--Step 0.bra t_extdnrm.global setoxm1setoxm1:|--entry point for EXPM1(X), here X is finite, non-zero, non-NaN|--Step 1.|--Step 1.1movel (%a0),%d0 | ...load part of input Xandil #0x7FFF0000,%d0 | ...biased expo. of Xcmpil #0x3FFD0000,%d0 | ...1/4bges EM1CON1 | ...|X| >= 1/4bra EM1SMEM1CON1:|--Step 1.3|--The case |X| >= 1/4movew 4(%a0),%d0 | ...expo. and partial sig. of |X|cmpil #0x4004C215,%d0 | ...70log2 rounded up to 16 bitsbles EM1MAIN | ...1/4 <= |X| <= 70log2bra EM1BIGEM1MAIN:|--Step 2.|--This is the case: 1/4 <= |X| <= 70 log2.fmovex (%a0),%fp0 | ...load input from (a0)fmovex %fp0,%fp1fmuls #0x42B8AA3B,%fp0 | ...64/log2 * Xfmovemx %fp2-%fp2/%fp3,-(%a7) | ...save fp2| MOVE.W #$3F81,EM1A4 ...prefetch in CB modefmovel %fp0,%d0 | ...N = int( X * 64/log2 )lea EXPTBL,%a1fmovel %d0,%fp0 | ...convert to floating-formatmovel %d0,L_SCR1(%a6) | ...save N temporarilyandil #0x3F,%d0 | ...D0 is J = N mod 64lsll #4,%d0addal %d0,%a1 | ...address of 2^(J/64)movel L_SCR1(%a6),%d0asrl #6,%d0 | ...D0 is Mmovel %d0,L_SCR1(%a6) | ...save a copy of M| MOVE.W #$3FDC,L2 ...prefetch L2 in CB mode|--Step 3.|--fp1,fp2 saved on the stack. fp0 is N, fp1 is X,|--a0 points to 2^(J/64), D0 and a1 both contain Mfmovex %fp0,%fp2fmuls #0xBC317218,%fp0 | ...N * L1, L1 = lead(-log2/64)fmulx L2,%fp2 | ...N * L2, L1+L2 = -log2/64faddx %fp1,%fp0 | ...X + N*L1faddx %fp2,%fp0 | ...fp0 is R, reduced arg.| MOVE.W #$3FC5,EM1A2 ...load EM1A2 in cacheaddiw #0x3FFF,%d0 | ...D0 is biased expo. of 2^M|--Step 4.|--WE NOW COMPUTE EXP(R)-1 BY A POLYNOMIAL|-- R + R*R*(A1 + R*(A2 + R*(A3 + R*(A4 + R*(A5 + R*A6)))))|--TO FULLY UTILIZE THE PIPELINE, WE COMPUTE S = R*R|--[R*S*(A2+S*(A4+S*A6))] + [R+S*(A1+S*(A3+S*A5))]fmovex %fp0,%fp1fmulx %fp1,%fp1 | ...fp1 IS S = R*Rfmoves #0x3950097B,%fp2 | ...fp2 IS a6| MOVE.W #0,2(%a1) ...load 2^(J/64) in cachefmulx %fp1,%fp2 | ...fp2 IS S*A6fmovex %fp1,%fp3fmuls #0x3AB60B6A,%fp3 | ...fp3 IS S*A5faddd EM1A4,%fp2 | ...fp2 IS A4+S*A6faddd EM1A3,%fp3 | ...fp3 IS A3+S*A5movew %d0,SC(%a6) | ...SC is 2^(M) in extendedclrw SC+2(%a6)movel #0x80000000,SC+4(%a6)clrl SC+8(%a6)fmulx %fp1,%fp2 | ...fp2 IS S*(A4+S*A6)movel L_SCR1(%a6),%d0 | ...D0 is Mnegw %d0 | ...D0 is -Mfmulx %fp1,%fp3 | ...fp3 IS S*(A3+S*A5)addiw #0x3FFF,%d0 | ...biased expo. of 2^(-M)faddd EM1A2,%fp2 | ...fp2 IS A2+S*(A4+S*A6)fadds #0x3F000000,%fp3 | ...fp3 IS A1+S*(A3+S*A5)fmulx %fp1,%fp2 | ...fp2 IS S*(A2+S*(A4+S*A6))oriw #0x8000,%d0 | ...signed/expo. of -2^(-M)movew %d0,ONEBYSC(%a6) | ...OnebySc is -2^(-M)clrw ONEBYSC+2(%a6)movel #0x80000000,ONEBYSC+4(%a6)clrl ONEBYSC+8(%a6)fmulx %fp3,%fp1 | ...fp1 IS S*(A1+S*(A3+S*A5))| ...fp3 releasedfmulx %fp0,%fp2 | ...fp2 IS R*S*(A2+S*(A4+S*A6))faddx %fp1,%fp0 | ...fp0 IS R+S*(A1+S*(A3+S*A5))| ...fp1 releasedfaddx %fp2,%fp0 | ...fp0 IS EXP(R)-1| ...fp2 releasedfmovemx (%a7)+,%fp2-%fp2/%fp3 | ...fp2 restored|--Step 5|--Compute 2^(J/64)*pfmulx (%a1),%fp0 | ...2^(J/64)*(Exp(R)-1)|--Step 6|--Step 6.1movel L_SCR1(%a6),%d0 | ...retrieve Mcmpil #63,%d0bles MLE63|--Step 6.2 M >= 64fmoves 12(%a1),%fp1 | ...fp1 is tfaddx ONEBYSC(%a6),%fp1 | ...fp1 is t+OnebyScfaddx %fp1,%fp0 | ...p+(t+OnebySc), fp1 releasedfaddx (%a1),%fp0 | ...T+(p+(t+OnebySc))bras EM1SCALEMLE63:|--Step 6.3 M <= 63cmpil #-3,%d0bges MGEN3MLTN3:|--Step 6.4 M <= -4fadds 12(%a1),%fp0 | ...p+tfaddx (%a1),%fp0 | ...T+(p+t)faddx ONEBYSC(%a6),%fp0 | ...OnebySc + (T+(p+t))bras EM1SCALEMGEN3:|--Step 6.5 -3 <= M <= 63fmovex (%a1)+,%fp1 | ...fp1 is Tfadds (%a1),%fp0 | ...fp0 is p+tfaddx ONEBYSC(%a6),%fp1 | ...fp1 is T+OnebyScfaddx %fp1,%fp0 | ...(T+OnebySc)+(p+t)EM1SCALE:|--Step 6.6fmovel %d1,%FPCRfmulx SC(%a6),%fp0bra t_frcinxEM1SM:|--Step 7 |X| < 1/4.cmpil #0x3FBE0000,%d0 | ...2^(-65)bges EM1POLYEM1TINY:|--Step 8 |X| < 2^(-65)cmpil #0x00330000,%d0 | ...2^(-16312)blts EM12TINY|--Step 8.2movel #0x80010000,SC(%a6) | ...SC is -2^(-16382)movel #0x80000000,SC+4(%a6)clrl SC+8(%a6)fmovex (%a0),%fp0fmovel %d1,%FPCRfaddx SC(%a6),%fp0bra t_frcinxEM12TINY:|--Step 8.3fmovex (%a0),%fp0fmuld TWO140,%fp0movel #0x80010000,SC(%a6)movel #0x80000000,SC+4(%a6)clrl SC+8(%a6)faddx SC(%a6),%fp0fmovel %d1,%FPCRfmuld TWON140,%fp0bra t_frcinxEM1POLY:|--Step 9 exp(X)-1 by a simple polynomialfmovex (%a0),%fp0 | ...fp0 is Xfmulx %fp0,%fp0 | ...fp0 is S := X*Xfmovemx %fp2-%fp2/%fp3,-(%a7) | ...save fp2fmoves #0x2F30CAA8,%fp1 | ...fp1 is B12fmulx %fp0,%fp1 | ...fp1 is S*B12fmoves #0x310F8290,%fp2 | ...fp2 is B11fadds #0x32D73220,%fp1 | ...fp1 is B10+S*B12fmulx %fp0,%fp2 | ...fp2 is S*B11fmulx %fp0,%fp1 | ...fp1 is S*(B10 + ...fadds #0x3493F281,%fp2 | ...fp2 is B9+S*...faddd EM1B8,%fp1 | ...fp1 is B8+S*...fmulx %fp0,%fp2 | ...fp2 is S*(B9+...fmulx %fp0,%fp1 | ...fp1 is S*(B8+...faddd EM1B7,%fp2 | ...fp2 is B7+S*...faddd EM1B6,%fp1 | ...fp1 is B6+S*...fmulx %fp0,%fp2 | ...fp2 is S*(B7+...fmulx %fp0,%fp1 | ...fp1 is S*(B6+...faddd EM1B5,%fp2 | ...fp2 is B5+S*...faddd EM1B4,%fp1 | ...fp1 is B4+S*...fmulx %fp0,%fp2 | ...fp2 is S*(B5+...fmulx %fp0,%fp1 | ...fp1 is S*(B4+...faddd EM1B3,%fp2 | ...fp2 is B3+S*...faddx EM1B2,%fp1 | ...fp1 is B2+S*...fmulx %fp0,%fp2 | ...fp2 is S*(B3+...fmulx %fp0,%fp1 | ...fp1 is S*(B2+...fmulx %fp0,%fp2 | ...fp2 is S*S*(B3+...)fmulx (%a0),%fp1 | ...fp1 is X*S*(B2...fmuls #0x3F000000,%fp0 | ...fp0 is S*B1faddx %fp2,%fp1 | ...fp1 is Q| ...fp2 releasedfmovemx (%a7)+,%fp2-%fp2/%fp3 | ...fp2 restoredfaddx %fp1,%fp0 | ...fp0 is S*B1+Q| ...fp1 releasedfmovel %d1,%FPCRfaddx (%a0),%fp0bra t_frcinxEM1BIG:|--Step 10 |X| > 70 log2movel (%a0),%d0cmpil #0,%d0bgt EXPC1|--Step 10.2fmoves #0xBF800000,%fp0 | ...fp0 is -1fmovel %d1,%FPCRfadds #0x00800000,%fp0 | ...-1 + 2^(-126)bra t_frcinx|end
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