diff options
Diffstat (limited to 'arch/m68k/fpsp040/setox.S')
| -rw-r--r-- | arch/m68k/fpsp040/setox.S | 865 |
1 files changed, 865 insertions, 0 deletions
diff --git a/arch/m68k/fpsp040/setox.S b/arch/m68k/fpsp040/setox.S new file mode 100644 index 00000000000..0aa75f9bf7d --- /dev/null +++ b/arch/m68k/fpsp040/setox.S @@ -0,0 +1,865 @@ +| +| 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 |