| | |

| 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 |