blob: 0aa75f9bf7d1872f0b47ffa2b64efe585f26cb98 [file] [log] [blame]
 | | 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 0x3FDC0000,0x82E30865,0x4361C4C6,0x00000000 EXPA3: .long 0x3FA55555,0x55554431 EXPA2: .long 0x3FC55555,0x55554018 HUGE: .long 0x7FFE0000,0xFFFFFFFF,0xFFFFFFFF,0x00000000 TINY: .long 0x00010000,0xFFFFFFFF,0xFFFFFFFF,0x00000000 EM1A4: .long 0x3F811111,0x11174385 EM1A3: .long 0x3FA55555,0x55554F5A EM1A2: .long 0x3FC55555,0x55555555,0x00000000,0x00000000 EM1B8: .long 0x3EC71DE3,0xA5774682 EM1B7: .long 0x3EFA01A0,0x19D7CB68 EM1B6: .long 0x3F2A01A0,0x1A019DF3 EM1B5: .long 0x3F56C16C,0x16C170E2 EM1B4: .long 0x3F811111,0x11111111 EM1B3: .long 0x3FA55555,0x55555555 EM1B2: .long 0x3FFC0000,0xAAAAAAAA,0xAAAAAAAB .long 0x00000000 TWO140: .long 0x48B00000,0x00000000 TWON140: .long 0x37300000,0x00000000 EXPTBL: .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 setoxd setoxd: |--entry point for EXP(X), X is denormalized movel (%a0),%d0 andil #0x80000000,%d0 oril #0x00800000,%d0 | ...sign(X)*2^(-126) movel %d0,-(%sp) fmoves #0x3F800000,%fp0 fmovel %d1,%fpcr fadds (%sp)+,%fp0 bra t_frcinx .global setox setox: |--entry point for EXP(X), here X is finite, non-zero, and not NaN's |--Step 1. movel (%a0),%d0 | ...load part of input X andil #0x7FFF0000,%d0 | ...biased expo. of X cmpil #0x3FBE0000,%d0 | ...2^(-65) bges EXPC1 | ...normal case bra EXPSM EXPC1: |--The case |X| >= 2^(-65) movew 4(%a0),%d0 | ...expo. and partial sig. of |X| cmpil #0x400CB167,%d0 | ...16380 log2 trunc. 16 bits blts EXPMAIN | ...normal case bra EXPBIG EXPMAIN: |--Step 2. |--This is the normal branch: 2^(-65) <= |X| < 16380 log2. fmovex (%a0),%fp0 | ...load input from (a0) fmovex %fp0,%fp1 fmuls #0x42B8AA3B,%fp0 | ...64/log2 * X fmovemx %fp2-%fp2/%fp3,-(%a7) | ...save fp2 movel #0,ADJFLAG(%a6) fmovel %fp0,%d0 | ...N = int( X * 64/log2 ) lea EXPTBL,%a1 fmovel %d0,%fp0 | ...convert to floating-format movel %d0,L_SCR1(%a6) | ...save N temporarily andil #0x3F,%d0 | ...D0 is J = N mod 64 lsll #4,%d0 addal %d0,%a1 | ...address of 2^(J/64) movel L_SCR1(%a6),%d0 asrl #6,%d0 | ...D0 is M addiw #0x3FFF,%d0 | ...biased expo. of 2^(M) movew L2,L_SCR1(%a6) | ...prefetch L2, no need in CB EXPCONT1: |--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,%fp2 fmuls #0xBC317218,%fp0 | ...N * L1, L1 = lead(-log2/64) fmulx L2,%fp2 | ...N * L2, L1+L2 = -log2/64 faddx %fp1,%fp0 | ...X + N*L1 faddx %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,%fp1 fmulx %fp1,%fp1 | ...fp1 IS S = R*R fmoves #0x3AB60B70,%fp2 | ...fp2 IS A5 | MOVE.W #0,2(%a1) ...load 2^(J/64) in cache fmulx %fp1,%fp2 | ...fp2 IS S*A5 fmovex %fp1,%fp3 fmuls #0x3C088895,%fp3 | ...fp3 IS S*A4 faddd EXPA3,%fp2 | ...fp2 IS A3+S*A5 faddd EXPA2,%fp3 | ...fp3 IS A2+S*A4 fmulx %fp1,%fp2 | ...fp2 IS S*(A3+S*A5) movew %d0,SCALE(%a6) | ...SCALE is 2^(M) in extended clrw 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 released fmovex (%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 restored fadds (%a1),%fp0 | ...accurate 2^(J/64) faddx %fp1,%fp0 | ...2^(J/64) + 2^(J/64)*... movel ADJFLAG(%a6),%d0 |--Step 6 tstl %d0 beqs NORMAL ADJUST: fmulx ADJSCALE(%a6),%fp0 NORMAL: fmovel %d1,%FPCR | ...restore user FPCR fmulx SCALE(%a6),%fp0 | ...multiply 2^(M) bra t_frcinx EXPSM: |--Step 7 fmovemx (%a0),%fp0-%fp0 | ...in case X is denormalized fmovel %d1,%FPCR fadds #0x3F800000,%fp0 | ...1+X in user mode bra t_frcinx EXPBIG: |--Step 8 cmpil #0x400CB27C,%d0 | ...16480 log2 bgts EXP2BIG |--Steps 8.2 -- 8.6 fmovex (%a0),%fp0 | ...load input from (a0) fmovex %fp0,%fp1 fmuls #0x42B8AA3B,%fp0 | ...64/log2 * X fmovemx %fp2-%fp2/%fp3,-(%a7) | ...save fp2 movel #1,ADJFLAG(%a6) fmovel %fp0,%d0 | ...N = int( X * 64/log2 ) lea EXPTBL,%a1 fmovel %d0,%fp0 | ...convert to floating-format movel %d0,L_SCR1(%a6) | ...save N temporarily andil #0x3F,%d0 | ...D0 is J = N mod 64 lsll #4,%d0 addal %d0,%a1 | ...address of 2^(J/64) movel L_SCR1(%a6),%d0 asrl #6,%d0 | ...D0 is K movel %d0,L_SCR1(%a6) | ...save K temporarily asrl #1,%d0 | ...D0 is M1 subl %d0,L_SCR1(%a6) | ...a1 is M addiw #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 M addiw #0x3FFF,%d0 | ...biased expo. of 2^(M) bra EXPCONT1 | ...go back to Step 3 EXP2BIG: |--Step 9 fmovel %d1,%FPCR movel (%a0),%d0 bclrb #sign_bit,(%a0) | ...setox always returns positive cmpil #0,%d0 blt t_unfl bra t_ovfl .global setoxm1d setoxm1d: |--entry point for EXPM1(X), here X is denormalized |--Step 0. bra t_extdnrm .global setoxm1 setoxm1: |--entry point for EXPM1(X), here X is finite, non-zero, non-NaN |--Step 1. |--Step 1.1 movel (%a0),%d0 | ...load part of input X andil #0x7FFF0000,%d0 | ...biased expo. of X cmpil #0x3FFD0000,%d0 | ...1/4 bges EM1CON1 | ...|X| >= 1/4 bra EM1SM EM1CON1: |--Step 1.3 |--The case |X| >= 1/4 movew 4(%a0),%d0 | ...expo. and partial sig. of |X| cmpil #0x4004C215,%d0 | ...70log2 rounded up to 16 bits bles EM1MAIN | ...1/4 <= |X| <= 70log2 bra EM1BIG EM1MAIN: |--Step 2. |--This is the case: 1/4 <= |X| <= 70 log2. fmovex (%a0),%fp0 | ...load input from (a0) fmovex %fp0,%fp1 fmuls #0x42B8AA3B,%fp0 | ...64/log2 * X fmovemx %fp2-%fp2/%fp3,-(%a7) | ...save fp2 | MOVE.W #\$3F81,EM1A4 ...prefetch in CB mode fmovel %fp0,%d0 | ...N = int( X * 64/log2 ) lea EXPTBL,%a1 fmovel %d0,%fp0 | ...convert to floating-format movel %d0,L_SCR1(%a6) | ...save N temporarily andil #0x3F,%d0 | ...D0 is J = N mod 64 lsll #4,%d0 addal %d0,%a1 | ...address of 2^(J/64) movel L_SCR1(%a6),%d0 asrl #6,%d0 | ...D0 is M movel %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 M fmovex %fp0,%fp2 fmuls #0xBC317218,%fp0 | ...N * L1, L1 = lead(-log2/64) fmulx L2,%fp2 | ...N * L2, L1+L2 = -log2/64 faddx %fp1,%fp0 | ...X + N*L1 faddx %fp2,%fp0 | ...fp0 is R, reduced arg. | MOVE.W #\$3FC5,EM1A2 ...load EM1A2 in cache addiw #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,%fp1 fmulx %fp1,%fp1 | ...fp1 IS S = R*R fmoves #0x3950097B,%fp2 | ...fp2 IS a6 | MOVE.W #0,2(%a1) ...load 2^(J/64) in cache fmulx %fp1,%fp2 | ...fp2 IS S*A6 fmovex %fp1,%fp3 fmuls #0x3AB60B6A,%fp3 | ...fp3 IS S*A5 faddd EM1A4,%fp2 | ...fp2 IS A4+S*A6 faddd EM1A3,%fp3 | ...fp3 IS A3+S*A5 movew %d0,SC(%a6) | ...SC is 2^(M) in extended clrw 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 M negw %d0 | ...D0 is -M fmulx %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 released fmulx %fp0,%fp2 | ...fp2 IS R*S*(A2+S*(A4+S*A6)) faddx %fp1,%fp0 | ...fp0 IS R+S*(A1+S*(A3+S*A5)) | ...fp1 released faddx %fp2,%fp0 | ...fp0 IS EXP(R)-1 | ...fp2 released fmovemx (%a7)+,%fp2-%fp2/%fp3 | ...fp2 restored |--Step 5 |--Compute 2^(J/64)*p fmulx (%a1),%fp0 | ...2^(J/64)*(Exp(R)-1) |--Step 6 |--Step 6.1 movel L_SCR1(%a6),%d0 | ...retrieve M cmpil #63,%d0 bles MLE63 |--Step 6.2 M >= 64 fmoves 12(%a1),%fp1 | ...fp1 is t faddx ONEBYSC(%a6),%fp1 | ...fp1 is t+OnebySc faddx %fp1,%fp0 | ...p+(t+OnebySc), fp1 released faddx (%a1),%fp0 | ...T+(p+(t+OnebySc)) bras EM1SCALE MLE63: |--Step 6.3 M <= 63 cmpil #-3,%d0 bges MGEN3 MLTN3: |--Step 6.4 M <= -4 fadds 12(%a1),%fp0 | ...p+t faddx (%a1),%fp0 | ...T+(p+t) faddx ONEBYSC(%a6),%fp0 | ...OnebySc + (T+(p+t)) bras EM1SCALE MGEN3: |--Step 6.5 -3 <= M <= 63 fmovex (%a1)+,%fp1 | ...fp1 is T fadds (%a1),%fp0 | ...fp0 is p+t faddx ONEBYSC(%a6),%fp1 | ...fp1 is T+OnebySc faddx %fp1,%fp0 | ...(T+OnebySc)+(p+t) EM1SCALE: |--Step 6.6 fmovel %d1,%FPCR fmulx SC(%a6),%fp0 bra t_frcinx EM1SM: |--Step 7 |X| < 1/4. cmpil #0x3FBE0000,%d0 | ...2^(-65) bges EM1POLY EM1TINY: |--Step 8 |X| < 2^(-65) cmpil #0x00330000,%d0 | ...2^(-16312) blts EM12TINY |--Step 8.2 movel #0x80010000,SC(%a6) | ...SC is -2^(-16382) movel #0x80000000,SC+4(%a6) clrl SC+8(%a6) fmovex (%a0),%fp0 fmovel %d1,%FPCR faddx SC(%a6),%fp0 bra t_frcinx EM12TINY: |--Step 8.3 fmovex (%a0),%fp0 fmuld TWO140,%fp0 movel #0x80010000,SC(%a6) movel #0x80000000,SC+4(%a6) clrl SC+8(%a6) faddx SC(%a6),%fp0 fmovel %d1,%FPCR fmuld TWON140,%fp0 bra t_frcinx EM1POLY: |--Step 9 exp(X)-1 by a simple polynomial fmovex (%a0),%fp0 | ...fp0 is X fmulx %fp0,%fp0 | ...fp0 is S := X*X fmovemx %fp2-%fp2/%fp3,-(%a7) | ...save fp2 fmoves #0x2F30CAA8,%fp1 | ...fp1 is B12 fmulx %fp0,%fp1 | ...fp1 is S*B12 fmoves #0x310F8290,%fp2 | ...fp2 is B11 fadds #0x32D73220,%fp1 | ...fp1 is B10+S*B12 fmulx %fp0,%fp2 | ...fp2 is S*B11 fmulx %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*B1 faddx %fp2,%fp1 | ...fp1 is Q | ...fp2 released fmovemx (%a7)+,%fp2-%fp2/%fp3 | ...fp2 restored faddx %fp1,%fp0 | ...fp0 is S*B1+Q | ...fp1 released fmovel %d1,%FPCR faddx (%a0),%fp0 bra t_frcinx EM1BIG: |--Step 10 |X| > 70 log2 movel (%a0),%d0 cmpil #0,%d0 bgt EXPC1 |--Step 10.2 fmoves #0xBF800000,%fp0 | ...fp0 is -1 fmovel %d1,%FPCR fadds #0x00800000,%fp0 | ...-1 + 2^(-126) bra t_frcinx |end