diff options
Diffstat (limited to 'ANDROID_3.4.5/arch/m68k/fpsp040/setox.S')
| -rw-r--r-- | ANDROID_3.4.5/arch/m68k/fpsp040/setox.S | 864 |
1 files changed, 0 insertions, 864 deletions
diff --git a/ANDROID_3.4.5/arch/m68k/fpsp040/setox.S b/ANDROID_3.4.5/arch/m68k/fpsp040/setox.S deleted file mode 100644 index f1acf7e3..00000000 --- a/ANDROID_3.4.5/arch/m68k/fpsp040/setox.S +++ /dev/null @@ -1,864 +0,0 @@ -| -| 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 -| -| For details on the license for this file, please see the -| file, README, in this same directory. - -|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 |