865 lines
28 KiB
ArmAsm
865 lines
28 KiB
ArmAsm
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| setox.sa 3.1 12/10/90
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| The entry point setox computes the exponential of a value.
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| setoxd does the same except the input value is a denormalized
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| number. setoxm1 computes exp(X)-1, and setoxm1d computes
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| exp(X)-1 for denormalized X.
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| INPUT
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| -----
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| Double-extended value in memory location pointed to by address
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| register a0.
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| OUTPUT
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| ------
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| exp(X) or exp(X)-1 returned in floating-point register fp0.
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| ACCURACY and MONOTONICITY
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| -------------------------
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| The returned result is within 0.85 ulps in 64 significant bit, i.e.
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| within 0.5001 ulp to 53 bits if the result is subsequently rounded
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| to double precision. The result is provably monotonic in double
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| precision.
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| SPEED
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| -----
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| Two timings are measured, both in the copy-back mode. The
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| first one is measured when the function is invoked the first time
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| (so the instructions and data are not in cache), and the
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| second one is measured when the function is reinvoked at the same
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| input argument.
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| The program setox takes approximately 210/190 cycles for input
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| argument X whose magnitude is less than 16380 log2, which
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| is the usual situation. For the less common arguments,
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| depending on their values, the program may run faster or slower --
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| but no worse than 10% slower even in the extreme cases.
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| The program setoxm1 takes approximately ??? / ??? cycles for input
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| argument X, 0.25 <= |X| < 70log2. For |X| < 0.25, it takes
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| approximately ??? / ??? cycles. For the less common arguments,
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| depending on their values, the program may run faster or slower --
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| but no worse than 10% slower even in the extreme cases.
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| ALGORITHM and IMPLEMENTATION NOTES
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| ----------------------------------
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| setoxd
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| ------
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| Step 1. Set ans := 1.0
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| Step 2. Return ans := ans + sign(X)*2^(-126). Exit.
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| Notes: This will always generate one exception -- inexact.
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| setox
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| -----
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| Step 1. Filter out extreme cases of input argument.
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| 1.1 If |X| >= 2^(-65), go to Step 1.3.
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| 1.2 Go to Step 7.
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| 1.3 If |X| < 16380 log(2), go to Step 2.
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| 1.4 Go to Step 8.
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| Notes: The usual case should take the branches 1.1 -> 1.3 -> 2.
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| To avoid the use of floating-point comparisons, a
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| compact representation of |X| is used. This format is a
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| 32-bit integer, the upper (more significant) 16 bits are
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| the sign and biased exponent field of |X|; the lower 16
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| bits are the 16 most significant fraction (including the
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| explicit bit) bits of |X|. Consequently, the comparisons
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| in Steps 1.1 and 1.3 can be performed by integer comparison.
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| Note also that the constant 16380 log(2) used in Step 1.3
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| is also in the compact form. Thus taking the branch
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| to Step 2 guarantees |X| < 16380 log(2). There is no harm
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| to have a small number of cases where |X| is less than,
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| but close to, 16380 log(2) and the branch to Step 9 is
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| taken.
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| Step 2. Calculate N = round-to-nearest-int( X * 64/log2 ).
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| 2.1 Set AdjFlag := 0 (indicates the branch 1.3 -> 2 was taken)
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| 2.2 N := round-to-nearest-integer( X * 64/log2 ).
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| 2.3 Calculate J = N mod 64; so J = 0,1,2,..., or 63.
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| 2.4 Calculate M = (N - J)/64; so N = 64M + J.
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| 2.5 Calculate the address of the stored value of 2^(J/64).
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| 2.6 Create the value Scale = 2^M.
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| Notes: The calculation in 2.2 is really performed by
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| Z := X * constant
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| N := round-to-nearest-integer(Z)
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| where
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| constant := single-precision( 64/log 2 ).
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| Using a single-precision constant avoids memory access.
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| Another effect of using a single-precision "constant" is
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| that the calculated value Z is
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| Z = X*(64/log2)*(1+eps), |eps| <= 2^(-24).
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| This error has to be considered later in Steps 3 and 4.
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| Step 3. Calculate X - N*log2/64.
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| 3.1 R := X + N*L1, where L1 := single-precision(-log2/64).
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| 3.2 R := R + N*L2, L2 := extended-precision(-log2/64 - L1).
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| Notes: a) The way L1 and L2 are chosen ensures L1+L2 approximate
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| the value -log2/64 to 88 bits of accuracy.
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| b) N*L1 is exact because N is no longer than 22 bits and
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| L1 is no longer than 24 bits.
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| c) The calculation X+N*L1 is also exact due to cancellation.
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| Thus, R is practically X+N(L1+L2) to full 64 bits.
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| d) It is important to estimate how large can |R| be after
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| Step 3.2.
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| N = rnd-to-int( X*64/log2 (1+eps) ), |eps|<=2^(-24)
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| X*64/log2 (1+eps) = N + f, |f| <= 0.5
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| X*64/log2 - N = f - eps*X 64/log2
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| X - N*log2/64 = f*log2/64 - eps*X
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| Now |X| <= 16446 log2, thus
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| |X - N*log2/64| <= (0.5 + 16446/2^(18))*log2/64
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| <= 0.57 log2/64.
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| This bound will be used in Step 4.
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| Step 4. Approximate exp(R)-1 by a polynomial
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| p = R + R*R*(A1 + R*(A2 + R*(A3 + R*(A4 + R*A5))))
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| Notes: a) In order to reduce memory access, the coefficients are
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| made as "short" as possible: A1 (which is 1/2), A4 and A5
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| are single precision; A2 and A3 are double precision.
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| b) Even with the restrictions above,
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| |p - (exp(R)-1)| < 2^(-68.8) for all |R| <= 0.0062.
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| Note that 0.0062 is slightly bigger than 0.57 log2/64.
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| c) To fully utilize the pipeline, p is separated into
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| two independent pieces of roughly equal complexities
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| p = [ R + R*S*(A2 + S*A4) ] +
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| [ S*(A1 + S*(A3 + S*A5)) ]
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| where S = R*R.
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| Step 5. Compute 2^(J/64)*exp(R) = 2^(J/64)*(1+p) by
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| ans := T + ( T*p + t)
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| where T and t are the stored values for 2^(J/64).
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| Notes: 2^(J/64) is stored as T and t where T+t approximates
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| 2^(J/64) to roughly 85 bits; T is in extended precision
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| and t is in single precision. Note also that T is rounded
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| to 62 bits so that the last two bits of T are zero. The
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| reason for such a special form is that T-1, T-2, and T-8
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| will all be exact --- a property that will give much
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| more accurate computation of the function EXPM1.
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| Step 6. Reconstruction of exp(X)
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| exp(X) = 2^M * 2^(J/64) * exp(R).
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| 6.1 If AdjFlag = 0, go to 6.3
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| 6.2 ans := ans * AdjScale
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| 6.3 Restore the user FPCR
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| 6.4 Return ans := ans * Scale. Exit.
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| Notes: If AdjFlag = 0, we have X = Mlog2 + Jlog2/64 + R,
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| |M| <= 16380, and Scale = 2^M. Moreover, exp(X) will
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| neither overflow nor underflow. If AdjFlag = 1, that
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| means that
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| X = (M1+M)log2 + Jlog2/64 + R, |M1+M| >= 16380.
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| Hence, exp(X) may overflow or underflow or neither.
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| When that is the case, AdjScale = 2^(M1) where M1 is
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| approximately M. Thus 6.2 will never cause over/underflow.
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| Possible exception in 6.4 is overflow or underflow.
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| The inexact exception is not generated in 6.4. Although
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| one can argue that the inexact flag should always be
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| raised, to simulate that exception cost to much than the
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| flag is worth in practical uses.
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| Step 7. Return 1 + X.
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| 7.1 ans := X
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| 7.2 Restore user FPCR.
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| 7.3 Return ans := 1 + ans. Exit
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| Notes: For non-zero X, the inexact exception will always be
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| raised by 7.3. That is the only exception raised by 7.3.
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| Note also that we use the FMOVEM instruction to move X
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| in Step 7.1 to avoid unnecessary trapping. (Although
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| the FMOVEM may not seem relevant since X is normalized,
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| the precaution will be useful in the library version of
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| this code where the separate entry for denormalized inputs
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| will be done away with.)
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| Step 8. Handle exp(X) where |X| >= 16380log2.
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| 8.1 If |X| > 16480 log2, go to Step 9.
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| (mimic 2.2 - 2.6)
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| 8.2 N := round-to-integer( X * 64/log2 )
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| 8.3 Calculate J = N mod 64, J = 0,1,...,63
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| 8.4 K := (N-J)/64, M1 := truncate(K/2), M = K-M1, AdjFlag := 1.
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| 8.5 Calculate the address of the stored value 2^(J/64).
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| 8.6 Create the values Scale = 2^M, AdjScale = 2^M1.
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| 8.7 Go to Step 3.
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| Notes: Refer to notes for 2.2 - 2.6.
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| Step 9. Handle exp(X), |X| > 16480 log2.
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| 9.1 If X < 0, go to 9.3
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| 9.2 ans := Huge, go to 9.4
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| 9.3 ans := Tiny.
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| 9.4 Restore user FPCR.
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| 9.5 Return ans := ans * ans. Exit.
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| Notes: Exp(X) will surely overflow or underflow, depending on
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| X's sign. "Huge" and "Tiny" are respectively large/tiny
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| extended-precision numbers whose square over/underflow
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| with an inexact result. Thus, 9.5 always raises the
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| inexact together with either overflow or underflow.
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| setoxm1d
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| --------
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| Step 1. Set ans := 0
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| Step 2. Return ans := X + ans. Exit.
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| Notes: This will return X with the appropriate rounding
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| precision prescribed by the user FPCR.
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| setoxm1
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| -------
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| Step 1. Check |X|
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| 1.1 If |X| >= 1/4, go to Step 1.3.
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| 1.2 Go to Step 7.
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| 1.3 If |X| < 70 log(2), go to Step 2.
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| 1.4 Go to Step 10.
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| Notes: The usual case should take the branches 1.1 -> 1.3 -> 2.
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| However, it is conceivable |X| can be small very often
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| because EXPM1 is intended to evaluate exp(X)-1 accurately
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| when |X| is small. For further details on the comparisons,
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| see the notes on Step 1 of setox.
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| Step 2. Calculate N = round-to-nearest-int( X * 64/log2 ).
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| 2.1 N := round-to-nearest-integer( X * 64/log2 ).
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| 2.2 Calculate J = N mod 64; so J = 0,1,2,..., or 63.
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| 2.3 Calculate M = (N - J)/64; so N = 64M + J.
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| 2.4 Calculate the address of the stored value of 2^(J/64).
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| 2.5 Create the values Sc = 2^M and OnebySc := -2^(-M).
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| Notes: See the notes on Step 2 of setox.
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| Step 3. Calculate X - N*log2/64.
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| 3.1 R := X + N*L1, where L1 := single-precision(-log2/64).
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| 3.2 R := R + N*L2, L2 := extended-precision(-log2/64 - L1).
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| Notes: Applying the analysis of Step 3 of setox in this case
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| shows that |R| <= 0.0055 (note that |X| <= 70 log2 in
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| this case).
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| Step 4. Approximate exp(R)-1 by a polynomial
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| p = R+R*R*(A1+R*(A2+R*(A3+R*(A4+R*(A5+R*A6)))))
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| Notes: a) In order to reduce memory access, the coefficients are
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| made as "short" as possible: A1 (which is 1/2), A5 and A6
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| are single precision; A2, A3 and A4 are double precision.
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| b) Even with the restriction above,
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| |p - (exp(R)-1)| < |R| * 2^(-72.7)
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| for all |R| <= 0.0055.
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| c) To fully utilize the pipeline, p is separated into
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| two independent pieces of roughly equal complexity
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| p = [ R*S*(A2 + S*(A4 + S*A6)) ] +
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| [ R + S*(A1 + S*(A3 + S*A5)) ]
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| where S = R*R.
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| Step 5. Compute 2^(J/64)*p by
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| p := T*p
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| where T and t are the stored values for 2^(J/64).
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| Notes: 2^(J/64) is stored as T and t where T+t approximates
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| 2^(J/64) to roughly 85 bits; T is in extended precision
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| and t is in single precision. Note also that T is rounded
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| to 62 bits so that the last two bits of T are zero. The
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| reason for such a special form is that T-1, T-2, and T-8
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| will all be exact --- a property that will be exploited
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| in Step 6 below. The total relative error in p is no
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| bigger than 2^(-67.7) compared to the final result.
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| Step 6. Reconstruction of exp(X)-1
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| exp(X)-1 = 2^M * ( 2^(J/64) + p - 2^(-M) ).
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| 6.1 If M <= 63, go to Step 6.3.
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| 6.2 ans := T + (p + (t + OnebySc)). Go to 6.6
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| 6.3 If M >= -3, go to 6.5.
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| 6.4 ans := (T + (p + t)) + OnebySc. Go to 6.6
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| 6.5 ans := (T + OnebySc) + (p + t).
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| 6.6 Restore user FPCR.
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| 6.7 Return ans := Sc * ans. Exit.
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| Notes: The various arrangements of the expressions give accurate
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| evaluations.
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| Step 7. exp(X)-1 for |X| < 1/4.
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| 7.1 If |X| >= 2^(-65), go to Step 9.
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| 7.2 Go to Step 8.
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| Step 8. Calculate exp(X)-1, |X| < 2^(-65).
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| 8.1 If |X| < 2^(-16312), goto 8.3
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| 8.2 Restore FPCR; return ans := X - 2^(-16382). Exit.
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| 8.3 X := X * 2^(140).
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| 8.4 Restore FPCR; ans := ans - 2^(-16382).
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| Return ans := ans*2^(140). Exit
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| Notes: The idea is to return "X - tiny" under the user
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| precision and rounding modes. To avoid unnecessary
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| inefficiency, we stay away from denormalized numbers the
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| best we can. For |X| >= 2^(-16312), the straightforward
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| 8.2 generates the inexact exception as the case warrants.
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| Step 9. Calculate exp(X)-1, |X| < 1/4, by a polynomial
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| p = X + X*X*(B1 + X*(B2 + ... + X*B12))
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| Notes: a) In order to reduce memory access, the coefficients are
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| made as "short" as possible: B1 (which is 1/2), B9 to B12
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| are single precision; B3 to B8 are double precision; and
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| B2 is double extended.
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| b) Even with the restriction above,
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| |p - (exp(X)-1)| < |X| 2^(-70.6)
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| for all |X| <= 0.251.
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| Note that 0.251 is slightly bigger than 1/4.
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| c) To fully preserve accuracy, the polynomial is computed
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| as X + ( S*B1 + Q ) where S = X*X and
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| Q = X*S*(B2 + X*(B3 + ... + X*B12))
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| d) To fully utilize the pipeline, Q is separated into
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| two independent pieces of roughly equal complexity
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| Q = [ X*S*(B2 + S*(B4 + ... + S*B12)) ] +
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| [ S*S*(B3 + S*(B5 + ... + S*B11)) ]
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| Step 10. Calculate exp(X)-1 for |X| >= 70 log 2.
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| 10.1 If X >= 70log2 , exp(X) - 1 = exp(X) for all practical
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| purposes. Therefore, go to Step 1 of setox.
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| 10.2 If X <= -70log2, exp(X) - 1 = -1 for all practical purposes.
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| ans := -1
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| Restore user FPCR
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| Return ans := ans + 2^(-126). Exit.
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| Notes: 10.2 will always create an inexact and return -1 + tiny
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| in the user rounding precision and mode.
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| Copyright (C) Motorola, Inc. 1990
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| All Rights Reserved
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| For details on the license for this file, please see the
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| file, README, in this same directory.
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|setox idnt 2,1 | Motorola 040 Floating Point Software Package
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|section 8
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#include "fpsp.h"
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L2: .long 0x3FDC0000,0x82E30865,0x4361C4C6,0x00000000
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EXPA3: .long 0x3FA55555,0x55554431
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EXPA2: .long 0x3FC55555,0x55554018
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HUGE: .long 0x7FFE0000,0xFFFFFFFF,0xFFFFFFFF,0x00000000
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TINY: .long 0x00010000,0xFFFFFFFF,0xFFFFFFFF,0x00000000
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EM1A4: .long 0x3F811111,0x11174385
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EM1A3: .long 0x3FA55555,0x55554F5A
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EM1A2: .long 0x3FC55555,0x55555555,0x00000000,0x00000000
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EM1B8: .long 0x3EC71DE3,0xA5774682
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EM1B7: .long 0x3EFA01A0,0x19D7CB68
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EM1B6: .long 0x3F2A01A0,0x1A019DF3
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EM1B5: .long 0x3F56C16C,0x16C170E2
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EM1B4: .long 0x3F811111,0x11111111
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EM1B3: .long 0x3FA55555,0x55555555
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EM1B2: .long 0x3FFC0000,0xAAAAAAAA,0xAAAAAAAB
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.long 0x00000000
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TWO140: .long 0x48B00000,0x00000000
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TWON140: .long 0x37300000,0x00000000
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EXPTBL:
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.long 0x3FFF0000,0x80000000,0x00000000,0x00000000
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.long 0x3FFF0000,0x8164D1F3,0xBC030774,0x9F841A9B
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.long 0x3FFF0000,0x82CD8698,0xAC2BA1D8,0x9FC1D5B9
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.long 0x3FFF0000,0x843A28C3,0xACDE4048,0xA0728369
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.long 0x3FFF0000,0x85AAC367,0xCC487B14,0x1FC5C95C
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.long 0x3FFF0000,0x871F6196,0x9E8D1010,0x1EE85C9F
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.long 0x3FFF0000,0x88980E80,0x92DA8528,0x9FA20729
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.long 0x3FFF0000,0x8A14D575,0x496EFD9C,0xA07BF9AF
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.long 0x3FFF0000,0x8B95C1E3,0xEA8BD6E8,0xA0020DCF
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.long 0x3FFF0000,0x8D1ADF5B,0x7E5BA9E4,0x205A63DA
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.long 0x3FFF0000,0x8EA4398B,0x45CD53C0,0x1EB70051
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.long 0x3FFF0000,0x9031DC43,0x1466B1DC,0x1F6EB029
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.long 0x3FFF0000,0x91C3D373,0xAB11C338,0xA0781494
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.long 0x3FFF0000,0x935A2B2F,0x13E6E92C,0x9EB319B0
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.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
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|
fadds 12(%a1),%fp0 | ...p+t
|
|
faddx (%a1),%fp0 | ...T+(p+t)
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|
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
|