4 // Copyright (c) 2000 - 2003, Intel Corporation
5 // All rights reserved.
7 // Contributed 2000 by the Intel Numerics Group, Intel Corporation
9 // Redistribution and use in source and binary forms, with or without
10 // modification, are permitted provided that the following conditions are
13 // * Redistributions of source code must retain the above copyright
14 // notice, this list of conditions and the following disclaimer.
16 // * Redistributions in binary form must reproduce the above copyright
17 // notice, this list of conditions and the following disclaimer in the
18 // documentation and/or other materials provided with the distribution.
20 // * The name of Intel Corporation may not be used to endorse or promote
21 // products derived from this software without specific prior written
24 // THIS SOFTWARE IS PROVIDED BY THE COPYRIGHT HOLDERS AND CONTRIBUTORS
25 // "AS IS" AND ANY EXPRESS OR IMPLIED WARRANTIES, INCLUDING, BUT NOT
26 // LIMITED TO, THE IMPLIED WARRANTIES OF MERCHANTABILITY AND FITNESS FOR
27 // A PARTICULAR PURPOSE ARE DISCLAIMED. IN NO EVENT SHALL INTEL OR ITS
28 // CONTRIBUTORS BE LIABLE FOR ANY DIRECT, INDIRECT, INCIDENTAL, SPECIAL,
29 // EXEMPLARY, OR CONSEQUENTIAL DAMAGES (INCLUDING, BUT NOT LIMITED TO,
30 // PROCUREMENT OF SUBSTITUTE GOODS OR SERVICES; LOSS OF USE, DATA, OR
31 // PROFITS; OR BUSINESS INTERRUPTION) HOWEVER CAUSED AND ON ANY THEORY
32 // OF LIABILITY, WHETHER IN CONTRACT, STRICT LIABILITY OR TORT (INCLUDING
33 // NEGLIGENCE OR OTHERWISE) ARISING IN ANY WAY OUT OF THE USE OF THIS
34 // SOFTWARE, EVEN IF ADVISED OF THE POSSIBILITY OF SUCH DAMAGE.
36 // Intel Corporation is the author of this code, and requests that all
37 // problem reports or change requests be submitted to it directly at
38 // http://www.intel.com/software/products/opensource/libraries/num.htm.
41 // 02/02/00 Initial Version
42 // 05/13/02 Rescheduled for speed, changed interface to pass
43 // parameters in fp registers
44 // 02/10/03 Reordered header: .section, .global, .proc, .align;
45 // used data8 for long double data storage
47 //*********************************************************************
48 //*********************************************************************
50 // Function: __libm_pi_by_two_reduce(x) return r, c, and N where
51 // x = N * pi/4 + (r+c) , where |r+c| <= pi/4.
52 // This function is not designed to be used by the
55 //*********************************************************************
57 // Accuracy: Returns double-precision values
59 //*********************************************************************
63 // Floating-Point Registers:
64 // f8 = Input x, return value r
65 // f9 = return value c
68 // General Purpose Registers:
69 // r8 = return value N
72 // Predicate Registers: p6-p14
74 //*********************************************************************
76 // IEEE Special Conditions:
78 // No condions should be raised.
80 //*********************************************************************
85 // For the forward trigonometric functions sin, cos, sincos, and
86 // tan, the original algorithms for IA 64 handle arguments up to
87 // 1 ulp less than 2^63 in magnitude. For double-extended arguments x,
88 // |x| >= 2^63, this routine returns N and r_hi, r_lo where
90 // x is accurately approximated by
91 // 2*K*pi + N * pi/2 + r_hi + r_lo, |r_hi+r_lo| <= pi/4.
93 // CASE is 1 unless |r_hi + r_lo| < 2^(-33).
95 // The exact value of K is not determined, but that information is
96 // not required in trigonometric function computations.
98 // We first assume the argument x in question satisfies x >= 2^(63).
99 // In particular, it is positive. Negative x can be handled by symmetry:
101 // -x is accurately approximated by
102 // -2*K*pi + (-N) * pi/2 - (r_hi + r_lo), |r_hi+r_lo| <= pi/4.
104 // The idea of the reduction is that
106 // x * 2/pi = N_big + N + f, |f| <= 1/2
108 // Moreover, for double extended x, |f| >= 2^(-75). (This is an
109 // non-obvious fact found by enumeration using a special algorithm
110 // involving continued fraction.) The algorithm described below
111 // calculates N and an accurate approximation of f.
113 // Roughly speaking, an appropriate 256-bit (4 X 64) portion of
114 // 2/pi is multiplied with x to give the desired information.
116 // II. Representation of 2/PI
117 // ==========================
119 // The value of 2/pi in binary fixed-point is
121 // .101000101111100110......
123 // We store 2/pi in a table, starting at the position corresponding
124 // to bit position 63
126 // bit position 63 62 ... 0 -1 -2 -3 -4 -5 -6 -7 .... -16576
128 // 0 0 ... 0 . 1 0 1 0 1 0 1 .... X
131 // |__ implied binary pt
136 // This describes the algorithm in the most natural way using
137 // unsigned interger multiplication. The implementation section
138 // describes how the integer arithmetic is simulated.
140 // STEP 0. Initialization
141 // ----------------------
143 // Let the input argument x be
145 // x = 2^m * ( 1. b_1 b_2 b_3 ... b_63 ), 63 <= m <= 16383.
147 // The first crucial step is to fetch four 64-bit portions of 2/pi.
148 // To fulfill this goal, we calculate the bit position L of the
149 // beginning of these 256-bit quantity by
153 // Note that -16321 <= L <= -1 because 63 <= m <= 16383; and that
154 // the storage of 2/pi is adequate.
156 // Fetch P_1, P_2, P_3, P_4 beginning at bit position L thus:
158 // bit position L L-1 L-2 ... L-63
162 // each b can be 0 or 1. Also, let P_0 be the two bits correspoding to
163 // bit positions L+2 and L+1. So, when each of the P_j is interpreted
164 // with appropriate scaling, we have
166 // 2/pi = P_big + P_0 + (P_1 + P_2 + P_3 + P_4) + P_small
168 // Note that P_big and P_small can be ignored. The reasons are as follow.
169 // First, consider P_big. If P_big = 0, we can certainly ignore it.
170 // Otherwise, P_big >= 2^(L+3). Now,
172 // P_big * ulp(x) >= 2^(L+3) * 2^(m-63)
173 // >= 2^(65-m + m-63 )
176 // Thus, P_big * x is an integer of the form 4*K. So
178 // x = 4*K * (pi/2) + x*(P_0 + P_1 + P_2 + P_3 + P_4)*(pi/2)
179 // + x*P_small*(pi/2).
181 // Hence, P_big*x corresponds to information that can be ignored for
182 // trigonometic function evaluation.
184 // Next, we must estimate the effect of ignoring P_small. The absolute
185 // error made by ignoring P_small is bounded by
187 // |P_small * x| <= ulp(P_4) * x
188 // <= 2^(L-255) * 2^(m+1)
189 // <= 2^(62-m-255 + m + 1)
192 // Since for double-extended precision, x * 2/pi = integer + f,
193 // 0.5 >= |f| >= 2^(-75), the relative error introduced by ignoring
194 // P_small is bounded by 2^(-192+75) <= 2^(-117), which is acceptable.
196 // Further note that if x is split into x_hi + x_lo where x_lo is the
197 // two bits corresponding to bit positions 2^(m-62) and 2^(m-63); then
201 // is also an integer of the form 4*K; and thus can also be ignored.
202 // Let M := P_0 * x_lo which is a small integer. The main part of the
203 // calculation is really the multiplication of x with the four pieces
204 // P_1, P_2, P_3, and P_4.
206 // Unless the reduced argument is extremely small in magnitude, it
207 // suffices to carry out the multiplication of x with P_1, P_2, and
208 // P_3. x*P_4 will be carried out and added on as a correction only
209 // when it is found to be needed. Note also that x*P_4 need not be
210 // computed exactly. A straightforward multiplication suffices since
211 // the rounding error thus produced would be bounded by 2^(-3*64),
212 // that is 2^(-192) which is small enough as the reduced argument
213 // is bounded from below by 2^(-75).
215 // Now that we have four 64-bit data representing 2/pi and a
216 // 64-bit x. We first need to calculate a highly accurate product
217 // of x and P_1, P_2, P_3. This is best understood as integer
221 // STEP 1. Multiplication
222 // ----------------------
225 // --------- --------- ---------
226 // | P_1 | | P_2 | | P_3 |
227 // --------- --------- ---------
232 // ----------------------------------------------------
234 // --------- ---------
236 // --------- ---------
239 // --------- ---------
241 // --------- ---------
244 // --------- ---------
246 // --------- ---------
248 // ====================================================
249 // --------- --------- --------- ---------
250 // | S_0 | | S_1 | | S_2 | | S_3 |
251 // --------- --------- --------- ---------
255 // STEP 2. Get N and f
256 // -------------------
258 // Conceptually, after the individual pieces S_0, S_1, ..., are obtained,
259 // we have to sum them and obtain an integer part, N, and a fraction, f.
260 // Here, |f| <= 1/2, and N is an integer. Note also that N need only to
261 // be known to module 2^k, k >= 2. In the case when |f| is small enough,
262 // we would need to add in the value x*P_4.
265 // STEP 3. Get reduced argument
266 // ----------------------------
268 // The value f is not yet the reduced argument that we seek. The
271 // x * 2/pi = 4K + N + f
275 // x = 2*K*pi + N * pi/2 + f * (pi/2).
277 // Thus, the reduced argument is given by
279 // reduced argument = f * pi/2.
281 // This multiplication must be performed to extra precision.
283 // IV. Implementation
284 // ==================
286 // Step 0. Initialization
287 // ----------------------
289 // Set sgn_x := sign(x); x := |x|; x_lo := 2 lsb of x.
291 // In memory, 2/pi is stored contigously as
293 // 0x00000000 0x00000000 0xA2F....
295 // |__ implied binary bit
297 // Given x = 2^m * 1.xxxx...xxx; we calculate L := 62 - m. Thus
298 // -1 <= L <= -16321. We fetch from memory 5 integer pieces of data.
300 // P_0 is the two bits corresponding to bit positions L+2 and L+1
301 // P_1 is the 64-bit starting at bit position L
302 // P_2 is the 64-bit starting at bit position L-64
303 // P_3 is the 64-bit starting at bit position L-128
304 // P_4 is the 64-bit starting at bit position L-192
306 // For example, if m = 63, P_0 would be 0 and P_1 would look like
309 // If m = 65, P_0 would be the two msb of 0xA, thus, P_0 is 10 in binary.
310 // P_1 in binary would be 1 0 0 0 1 0 1 1 1 1 ....
312 // Step 1. Multiplication
313 // ----------------------
315 // At this point, P_1, P_2, P_3, P_4 are integers. They are
316 // supposed to be interpreted as
319 // 2^(L-63-64) * P_2;
320 // 2^(L-63-128) * P_3;
321 // 2^(L-63-192) * P_4;
323 // Since each of them need to be multiplied to x, we would scale
324 // both x and the P_j's by some convenient factors: scale each
325 // of P_j's up by 2^(63-L), and scale x down by 2^(L-63).
327 // p_1 := fcvt.xf ( P_1 )
328 // p_2 := fcvt.xf ( P_2 ) * 2^(-64)
329 // p_3 := fcvt.xf ( P_3 ) * 2^(-128)
330 // p_4 := fcvt.xf ( P_4 ) * 2^(-192)
331 // x := replace exponent of x by -1
332 // because 2^m * 1.xxxx...xxx * 2^(L-63)
333 // is 2^(-1) * 1.xxxx...xxx
335 // We are now faced with the task of computing the following
337 // --------- --------- ---------
338 // | P_1 | | P_2 | | P_3 |
339 // --------- --------- ---------
344 // ----------------------------------------------------
346 // --------- ---------
348 // --------- ---------
350 // --------- ---------
352 // --------- ---------
354 // --------- ---------
356 // --------- ---------
358 // ====================================================
359 // ----------- --------- --------- ---------
360 // | S_0 | | S_1 | | S_2 | | S_3 |
361 // ----------- --------- --------- ---------
363 // | |___ binary point
365 // |___ possibly one more bit
367 // Let FPSR3 be set to round towards zero with widest precision
368 // and exponent range. Unless an explicit FPSR is given,
369 // round-to-nearest with widest precision and exponent range is
372 // Define sigma_C := 2^63; sigma_B := 2^(-1); sigma_C := 2^(-65).
374 // Tmp_C := fmpy.fpsr3( x, p_1 );
375 // If Tmp_C >= sigma_C then
377 // C_lo := x*p_1 - C_hi ...fma, exact
379 // C_hi := fadd.fpsr3(sigma_C, Tmp_C) - sigma_C
380 // ...subtraction is exact, regardless
381 // ...of rounding direction
382 // C_lo := x*p_1 - C_hi ...fma, exact
385 // Tmp_B := fmpy.fpsr3( x, p_2 );
386 // If Tmp_B >= sigma_B then
388 // B_lo := x*p_2 - B_hi ...fma, exact
390 // B_hi := fadd.fpsr3(sigma_B, Tmp_B) - sigma_B
391 // ...subtraction is exact, regardless
392 // ...of rounding direction
393 // B_lo := x*p_2 - B_hi ...fma, exact
396 // Tmp_A := fmpy.fpsr3( x, p_3 );
397 // If Tmp_A >= sigma_A then
399 // A_lo := x*p_3 - A_hi ...fma, exact
401 // A_hi := fadd.fpsr3(sigma_A, Tmp_A) - sigma_A
402 // ...subtraction is exact, regardless
403 // ...of rounding direction
404 // A_lo := x*p_3 - A_hi ...fma, exact
407 // ...Note that C_hi is of integer value. We need only the
408 // ...last few bits. Thus we can ensure C_hi is never a big
409 // ...integer, freeing us from overflow worry.
411 // Tmp_C := fadd.fpsr3( C_hi, 2^(70) ) - 2^(70);
412 // ...Tmp_C is the upper portion of C_hi
413 // C_hi := C_hi - Tmp_C
414 // ...0 <= C_hi < 2^7
416 // Step 2. Get N and f
417 // -------------------
419 // At this point, we have all the components to obtain
420 // S_0, S_1, S_2, S_3 and thus N and f. We start by adding
421 // C_lo and B_hi. This sum together with C_hi gives a good
422 // estimation of N and f.
424 // A := fadd.fpsr3( B_hi, C_lo )
425 // B := max( B_hi, C_lo )
426 // b := min( B_hi, C_lo )
428 // a := (B - A) + b ...exact. Note that a is either 0
431 // N := round_to_nearest_integer_value( A );
432 // f := A - N; ...exact because lsb(A) >= 2^(-64)
433 // ...and |f| <= 1/2.
435 // f := f + a ...exact because a is 0 or 2^(-64);
436 // ...the msb of the sum is <= 1/2
437 // ...lsb >= 2^(-64).
439 // N := convert to integer format( C_hi + N );
443 // If sgn_x == 1 (that is original x was negative)
445 // ...this maintains N to be non-negative, but still
446 // ...equivalent to the (negated N) mod 4.
455 // s_lo := (f - s_hi) + g;
461 // A := fadd.fpsr3( A_hi, B_lo )
462 // B := max( A_hi, B_lo )
463 // b := min( A_hi, B_lo )
465 // a := (B - A) + b ...exact. Note that a is either 0
469 // f_lo := (f - f_hi) + A;
471 // ...f-f_hi is exact because either |f| >= |A|, in which
472 // ...case f-f_hi is clearly exact; or otherwise, 0<|f|<|A|
473 // ...means msb(f) <= msb(A) = 2^(-64) => |f| = 2^(-64).
474 // ...If f = 2^(-64), f-f_hi involves cancellation and is
475 // ...exact. If f = -2^(-64), then A + f is exact. Hence
476 // ...f-f_hi is -A exactly, giving f_lo = 0.
480 // If |f| >= 2^(-50) then
484 // f_lo := (f_lo + A_lo) + x*p_4
485 // s_hi := f_hi + f_lo
486 // s_lo := (f_hi - s_hi) + f_lo
491 // Step 3. Get reduced argument
492 // ----------------------------
494 // If sgn_x == 0 (that is original x is positive)
496 // D_hi := Pi_by_2_hi
497 // D_lo := Pi_by_2_lo
498 // ...load from table
502 // D_hi := neg_Pi_by_2_hi
503 // D_lo := neg_Pi_by_2_lo
504 // ...load from table
508 // r_lo := s_hi*D_hi - r_hi ...fma
509 // r_lo := (s_hi*D_lo + r_lo) + s_lo*D_hi
511 // Return N, r_hi, r_lo
592 LOCAL_OBJECT_START(Constants_Bits_of_2_by_pi)
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702 data8 0xDD9B8E67EF3392B8,0x17C99B5861BC57E1
703 data8 0xC68351103ED84871,0xDDDD1C2DA118AF46
704 data8 0x2C21D7F359987AD9,0xC0549EFA864FFC06
705 data8 0x56AE79E536228922,0xAD38DC9367AAE855
706 data8 0x3826829BE7CAA40D,0x51B133990ED7A948
707 data8 0x0569F0B265A7887F,0x974C8836D1F9B392
708 data8 0x214A827B21CF98DC,0x9F405547DC3A74E1
709 data8 0x42EB67DF9DFE5FD4,0x5EA4677B7AACBAA2
710 data8 0xF65523882B55BA41,0x086E59862A218347
711 data8 0x39E6E389D49EE540,0xFB49E956FFCA0F1C
712 data8 0x8A59C52BFA94C5C1,0xD3CFC50FAE5ADB86
713 data8 0xC5476243853B8621,0x94792C8761107B4C
714 data8 0x2A1A2C8012BF4390,0x2688893C78E4C4A8
715 data8 0x7BDBE5C23AC4EAF4,0x268A67F7BF920D2B
716 data8 0xA365B1933D0B7CBD,0xDC51A463DD27DDE1
717 data8 0x6919949A9529A828,0xCE68B4ED09209F44
718 data8 0xCA984E638270237C,0x7E32B90F8EF5A7E7
719 data8 0x561408F1212A9DB5,0x4D7E6F5119A5ABF9
720 data8 0xB5D6DF8261DD9602,0x36169F3AC4A1A283
721 data8 0x6DED727A8D39A9B8,0x825C326B5B2746ED
722 data8 0x34007700D255F4FC,0x4D59018071E0E13F
723 data8 0x89B295F364A8F1AE,0xA74B38FC4CEAB2BB
724 LOCAL_OBJECT_END(Constants_Bits_of_2_by_pi)
726 LOCAL_OBJECT_START(Constants_Bits_of_pi_by_2)
727 data8 0xC90FDAA22168C234,0x00003FFF
728 data8 0xC4C6628B80DC1CD1,0x00003FBF
729 LOCAL_OBJECT_END(Constants_Bits_of_pi_by_2)
732 .global __libm_pi_by_2_reduce#
733 .proc __libm_pi_by_2_reduce#
736 __libm_pi_by_2_reduce:
739 // Place the two-piece result r (r_hi) in f8 and c (r_lo) in f9
740 // N is returned in r8
743 alloc r34 = ar.pfs,2,34,0,0
744 fsetc.s3 0x00,0x7F // Set sf3 to round to zero, 82-bit prec, td, ftz
748 addl GR_BASE = @ltoff(Constants_Bits_of_2_by_pi#), gp
750 mov GR_BIASL63 = 0x1003E
756 // 0 0 0 0 0. 1 0 1 0
757 // M 0 1 2 .... 63, 64 65 ... 127, 128
758 // ---------------------------------------------
759 // Segment 0. 1 , 2 , 3
760 // START = M - 63 M = 128 becomes 65
761 // LENGTH1 = START & 0x3F 65 become position 1
762 // SEGMENT = shr(START,6) + 1 0 maps to 1, 64 maps to 2,
763 // LENGTH2 = 64 - LENGTH1
764 // Address_BASE = shladd(SEGMENT,3) + BASE
768 getf.exp GR_Exp_x = FR_input_X
769 ld8 GR_BASE = [GR_BASE]
770 mov GR_TEMP5 = 0x0FFFE
774 // Define sigma_C := 2^63; sigma_B := 2^(-1); sigma_A := 2^(-65).
776 getf.sig GR_x_lo = FR_input_X
777 mov GR_TEMP6 = 0x0FFBE
782 // Special Code for testing DE arguments
783 // movl GR_BIASL63 = 0x0000000000013FFE
784 // movl GR_x_lo = 0xFFFFFFFFFFFFFFFF
785 // setf.exp FR_X = GR_BIASL63
786 // setf.sig FR_ScaleP3 = GR_x_lo
787 // fmerge.se FR_X = FR_X,FR_ScaleP3
788 // Set sgn_x := sign(x); x := |x|; x_lo := 2 lsb of x.
789 // 2/pi is stored contigously as
790 // 0x00000000 0x00000000.0xA2F....
791 // M = EXP - BIAS ( M >= 63)
792 // Given x = 2^m * 1.xxxx...xxx; we calculate L := 62 - m.
793 // Thus -1 <= L <= -16321.
795 setf.exp FR_sigma_B = GR_TEMP5
796 setf.exp FR_sigma_A = GR_TEMP6
797 extr.u GR_M = GR_Exp_x,0,17
802 and GR_x_lo = 0x03,GR_x_lo
803 sub GR_START = GR_M,GR_BIASL63
804 add GR_BASE = 8,GR_BASE // To effectively add 1 to SEGMENT
809 and GR_LENGTH1 = 0x3F,GR_START
810 shr.u GR_SEGMENT = GR_START,6
816 shladd GR_BASE = GR_SEGMENT,3,GR_BASE
817 sub GR_LENGTH2 = 0x40,GR_LENGTH1
818 cmp.le p6,p7 = 0x2,GR_LENGTH1
822 // P_0 is the two bits corresponding to bit positions L+2 and L+1
823 // P_1 is the 64-bit starting at bit position L
824 // P_2 is the 64-bit starting at bit position L-64
825 // P_3 is the 64-bit starting at bit position L-128
826 // P_4 is the 64-bit starting at bit position L-192
827 // P_1 is made up of Alo and Bhi
828 // P_1 = deposit Alo, position 0, length2 into P_1,position length1
829 // deposit Bhi, position length2, length1 into P_1, position 0
830 // P_2 is made up of Blo and Chi
831 // P_2 = deposit Blo, position 0, length2 into P_2, position length1
832 // deposit Chi, position length2, length1 into P_2, position 0
833 // P_3 is made up of Clo and Dhi
834 // P_3 = deposit Clo, position 0, length2 into P_3, position length1
835 // deposit Dhi, position length2, length1 into P_3, position 0
836 // P_4 is made up of Clo and Dhi
837 // P_4 = deposit Dlo, position 0, length2 into P_4, position length1
838 // deposit Ehi, position length2, length1 into P_4, position 0
840 ld8 GR_A = [GR_BASE],8
841 fabs FR_X = FR_input_X
842 (p7) cmp.eq.unc p8,p9 = 0x1,GR_LENGTH1
846 // ld_64 A at Base and increment Base by 8
847 // ld_64 B at Base and increment Base by 8
848 // ld_64 C at Base and increment Base by 8
849 // ld_64 D at Base and increment Base by 8
850 // ld_64 E at Base and increment Base by 8
852 // ---------------------
853 // A, B, C, D, and E look like | length1 | length2 |
854 // ---------------------
857 ld8 GR_B = [GR_BASE],8
858 movl GR_rshf = 0x43e8000000000000 // 1.10000 2^63 for right shift N_fix
863 ld8 GR_C = [GR_BASE],8
865 (p8) extr.u GR_Temp = GR_A,63,1
870 // P_0 = deposit Ahi, position length2, 2 bit into P_0 at position 0.
872 ld8 GR_D = [GR_BASE],8
873 shl GR_TEMP1 = GR_A,GR_LENGTH1 // MM instruction
874 (p6) shr.u GR_P_0 = GR_A,GR_LENGTH2 // MM instruction
879 ld8 GR_E = [GR_BASE],-40
880 shl GR_TEMP2 = GR_B,GR_LENGTH1 // MM instruction
881 shr.u GR_P_1 = GR_B,GR_LENGTH2 // MM instruction
886 // Load 16 bit of ASUB from (Base_Address_of_A - 2)
891 // Deposit element 63 from Ahi and place in element 0 of P_0.
896 (p7) ld2 GR_ASUB = [GR_BASE],8
897 shl GR_TEMP3 = GR_C,GR_LENGTH1 // MM instruction
898 shr.u GR_P_2 = GR_C,GR_LENGTH2 // MM instruction
903 setf.d FR_RSHF = GR_rshf // Form right shift const 1.100 * 2^63
904 shl GR_TEMP4 = GR_D,GR_LENGTH1 // MM instruction
905 shr.u GR_P_3 = GR_D,GR_LENGTH2 // MM instruction
910 (p7) and GR_P_0 = 0x03,GR_ASUB
911 (p6) and GR_P_0 = 0x03,GR_P_0
912 shr.u GR_P_4 = GR_E,GR_LENGTH2 // MM instruction
918 or GR_P_1 = GR_P_1,GR_TEMP1
919 (p8) and GR_P_0 = 0x1,GR_P_0
924 setf.sig FR_p_1 = GR_P_1
925 or GR_P_2 = GR_P_2,GR_TEMP2
926 (p8) shladd GR_P_0 = GR_P_0,1,GR_Temp
931 setf.sig FR_p_2 = GR_P_2
932 or GR_P_3 = GR_P_3,GR_TEMP3
933 fmerge.se FR_X = FR_sigma_B,FR_X
938 setf.sig FR_p_3 = GR_P_3
939 or GR_P_4 = GR_P_4,GR_TEMP4
940 pmpy2.r GR_M = GR_P_0,GR_x_lo
944 // P_1, P_2, P_3, P_4 are integers. They should be
946 // 2^(L-63-64) * P_2;
947 // 2^(L-63-128) * P_3;
948 // 2^(L-63-192) * P_4;
949 // Since each of them need to be multiplied to x, we would scale
950 // both x and the P_j's by some convenient factors: scale each
951 // of P_j's up by 2^(63-L), and scale x down by 2^(L-63).
952 // p_1 := fcvt.xf ( P_1 )
953 // p_2 := fcvt.xf ( P_2 ) * 2^(-64)
954 // p_3 := fcvt.xf ( P_3 ) * 2^(-128)
955 // p_4 := fcvt.xf ( P_4 ) * 2^(-192)
956 // x= Set x's exp to -1 because 2^m*1.x...x *2^(L-63)=2^(-1)*1.x...xxx
957 // --------- --------- ---------
958 // | P_1 | | P_2 | | P_3 |
959 // --------- --------- ---------
963 // ----------------------------------------------------
964 // --------- ---------
966 // --------- ---------
967 // --------- ---------
969 // --------- ---------
970 // --------- ---------
972 // --------- ---------
973 // ====================================================
974 // ----------- --------- --------- ---------
975 // | S_0 | | S_1 | | S_2 | | S_3 |
976 // ----------- --------- --------- ---------
977 // | |___ binary point
978 // |___ possibly one more bit
980 // Let FPSR3 be set to round towards zero with widest precision
981 // and exponent range. Unless an explicit FPSR is given,
982 // round-to-nearest with widest precision and exponent range is
985 setf.sig FR_p_4 = GR_P_4
986 mov GR_TEMP1 = 0x0FFBF
992 setf.exp FR_ScaleP2 = GR_TEMP1
993 mov GR_TEMP2 = 0x0FF7F
999 setf.exp FR_ScaleP3 = GR_TEMP2
1000 mov GR_TEMP4 = 0x1003E
1006 setf.exp FR_sigma_C = GR_TEMP4
1007 mov GR_Temp = 0x0FFDE
1008 fcvt.xuf.s1 FR_p_1 = FR_p_1
1013 setf.exp FR_TWOM33 = GR_Temp
1014 fcvt.xuf.s1 FR_p_2 = FR_p_2
1021 fcvt.xuf.s1 FR_p_3 = FR_p_3
1028 fcvt.xuf.s1 FR_p_4 = FR_p_4
1033 // Tmp_C := fmpy.fpsr3( x, p_1 );
1034 // Tmp_B := fmpy.fpsr3( x, p_2 );
1035 // Tmp_A := fmpy.fpsr3( x, p_3 );
1036 // If Tmp_C >= sigma_C then
1038 // C_lo := x*p_1 - C_hi ...fma, exact
1040 // C_hi := fadd.fpsr3(sigma_C, Tmp_C) - sigma_C
1041 // C_lo := x*p_1 - C_hi ...fma, exact
1043 // If Tmp_B >= sigma_B then
1045 // B_lo := x*p_2 - B_hi ...fma, exact
1047 // B_hi := fadd.fpsr3(sigma_B, Tmp_B) - sigma_B
1048 // B_lo := x*p_2 - B_hi ...fma, exact
1050 // If Tmp_A >= sigma_A then
1052 // A_lo := x*p_3 - A_hi ...fma, exact
1054 // A_hi := fadd.fpsr3(sigma_A, Tmp_A) - sigma_A
1055 // Exact, regardless ...of rounding direction
1056 // A_lo := x*p_3 - A_hi ...fma, exact
1060 fmpy.s3 FR_Tmp_C = FR_X,FR_p_1
1066 mov GR_TEMP3 = 0x0FF3F
1067 fmpy.s1 FR_p_2 = FR_p_2,FR_ScaleP2
1073 setf.exp FR_ScaleP4 = GR_TEMP3
1074 mov GR_TEMP4 = 0x10045
1075 fmpy.s1 FR_p_3 = FR_p_3,FR_ScaleP3
1081 fadd.s3 FR_C_hi = FR_sigma_C,FR_Tmp_C // For Tmp_C < sigma_C case
1087 setf.exp FR_Tmp2_C = GR_TEMP4
1089 fmpy.s3 FR_Tmp_B = FR_X,FR_p_2
1094 addl GR_BASE = @ltoff(Constants_Bits_of_pi_by_2#), gp
1095 fcmp.ge.s1 p12, p9 = FR_Tmp_C,FR_sigma_C
1100 fmpy.s3 FR_Tmp_A = FR_X,FR_p_3
1106 ld8 GR_BASE = [GR_BASE]
1107 (p12) mov FR_C_hi = FR_Tmp_C
1112 (p9) fsub.s1 FR_C_hi = FR_C_hi,FR_sigma_C
1120 // Step 3. Get reduced argument
1121 // If sgn_x == 0 (that is original x is positive)
1122 // D_hi := Pi_by_2_hi
1123 // D_lo := Pi_by_2_lo
1126 // D_hi := neg_Pi_by_2_hi
1127 // D_lo := neg_Pi_by_2_lo
1133 fmpy.s1 FR_p_4 = FR_p_4,FR_ScaleP4
1138 fadd.s3 FR_B_hi = FR_sigma_B,FR_Tmp_B // For Tmp_B < sigma_B case
1145 fadd.s3 FR_A_hi = FR_sigma_A,FR_Tmp_A // For Tmp_A < sigma_A case
1152 fcmp.ge.s1 p13, p10 = FR_Tmp_B,FR_sigma_B
1157 fms.s1 FR_C_lo = FR_X,FR_p_1,FR_C_hi
1163 ldfe FR_D_hi = [GR_BASE],16
1164 fcmp.ge.s1 p14, p11 = FR_Tmp_A,FR_sigma_A
1170 ldfe FR_D_lo = [GR_BASE]
1171 (p13) mov FR_B_hi = FR_Tmp_B
1176 (p10) fsub.s1 FR_B_hi = FR_B_hi,FR_sigma_B
1183 (p14) mov FR_A_hi = FR_Tmp_A
1188 (p11) fsub.s1 FR_A_hi = FR_A_hi,FR_sigma_A
1193 // Note that C_hi is of integer value. We need only the
1194 // last few bits. Thus we can ensure C_hi is never a big
1195 // integer, freeing us from overflow worry.
1196 // Tmp_C := fadd.fpsr3( C_hi, 2^(70) ) - 2^(70);
1197 // Tmp_C is the upper portion of C_hi
1200 fadd.s3 FR_Tmp_C = FR_C_hi,FR_Tmp2_C
1201 tbit.z p12,p9 = GR_Exp_x, 17
1207 fms.s1 FR_B_lo = FR_X,FR_p_2,FR_B_hi
1212 fadd.s3 FR_A = FR_B_hi,FR_C_lo
1219 fms.s1 FR_A_lo = FR_X,FR_p_3,FR_A_hi
1226 fsub.s1 FR_Tmp_C = FR_Tmp_C,FR_Tmp2_C
1231 // *******************
1232 // Step 2. Get N and f
1233 // *******************
1234 // We have all the components to obtain
1235 // S_0, S_1, S_2, S_3 and thus N and f. We start by adding
1236 // C_lo and B_hi. This sum together with C_hi estimates
1238 // A := fadd.fpsr3( B_hi, C_lo )
1239 // B := max( B_hi, C_lo )
1240 // b := min( B_hi, C_lo )
1243 fmax.s1 FR_B = FR_B_hi,FR_C_lo
1248 // We use a right-shift trick to get the integer part of A into the rightmost
1249 // bits of the significand by adding 1.1000..00 * 2^63. This operation is good
1250 // if |A| < 2^61, which it is in this case. We are doing this to save a few
1251 // cycles over using fcvt.fx followed by fnorm. The second step of the trick
1252 // is to subtract the same constant to float the rounded integer into a fp reg.
1256 // N := round_to_nearest_integer_value( A );
1257 fma.s1 FR_N_fix = FR_A, f1, FR_RSHF
1264 fmin.s1 FR_b = FR_B_hi,FR_C_lo
1269 // C_hi := C_hi - Tmp_C ...0 <= C_hi < 2^7
1270 fsub.s1 FR_C_hi = FR_C_hi,FR_Tmp_C
1277 // a := (B - A) + b: Exact - note that a is either 0 or 2^(-64).
1278 fsub.s1 FR_a = FR_B,FR_A
1285 fms.s1 FR_N = FR_N_fix, f1, FR_RSHF
1292 fadd.s1 FR_a = FR_a,FR_b
1297 // f := A - N; Exact because lsb(A) >= 2^(-64) and |f| <= 1/2.
1298 // N := convert to integer format( C_hi + N );
1303 fsub.s1 FR_f = FR_A,FR_N
1308 fadd.s1 FR_N = FR_N,FR_C_hi
1315 (p9) fsub.s1 FR_D_hi = f0, FR_D_hi
1320 (p9) fsub.s1 FR_D_lo = f0, FR_D_lo
1327 fadd.s1 FR_g = FR_A_hi,FR_B_lo // For Case 1, g=A_hi+B_lo
1332 fadd.s3 FR_A = FR_A_hi,FR_B_lo // For Case 2, A=A_hi+B_lo w/ sf3
1338 mov GR_Temp = 0x0FFCD // For Case 2, exponent of 2^-50
1339 fmax.s1 FR_B = FR_A_hi,FR_B_lo // For Case 2, B=max(A_hi,B_lo)
1344 // f = f + a Exact because a is 0 or 2^(-64);
1345 // the msb of the sum is <= 1/2 and lsb >= 2^(-64).
1347 setf.exp FR_TWOM50 = GR_Temp // For Case 2, form 2^-50
1348 fcvt.fx.s1 FR_N = FR_N
1353 fadd.s1 FR_f = FR_f,FR_a
1360 fmin.s1 FR_b = FR_A_hi,FR_B_lo // For Case 2, b=min(A_hi,B_lo)
1367 fsub.s1 FR_a = FR_B,FR_A // For Case 2, a=B-A
1374 fadd.s1 FR_s_hi = FR_f,FR_g // For Case 1, s_hi=f+g
1379 fadd.s1 FR_f_hi = FR_A,FR_f // For Case 2, f_hi=A+f
1386 fabs FR_f_abs = FR_f
1392 getf.sig GR_N = FR_N
1393 fsetc.s3 0x7F,0x40 // Reset sf3 to user settings + td
1400 fsub.s1 FR_s_lo = FR_f,FR_s_hi // For Case 1, s_lo=f-s_hi
1405 fsub.s1 FR_f_lo = FR_f,FR_f_hi // For Case 2, f_lo=f-f_hi
1412 fmpy.s1 FR_r_hi = FR_s_hi,FR_D_hi // For Case 1, r_hi=s_hi*D_hi
1417 fadd.s1 FR_a = FR_a,FR_b // For Case 2, a=a+b
1423 // If sgn_x == 1 (that is original x was negative)
1425 // this maintains N to be non-negative, but still
1426 // equivalent to the (negated N) mod 4.
1429 add GR_N = GR_N,GR_M
1430 fcmp.ge.s1 p13, p10 = FR_f_abs,FR_TWOM33
1431 mov GR_Temp = 0x00400
1436 (p9) sub GR_N = GR_Temp,GR_N
1437 fadd.s1 FR_s_lo = FR_s_lo,FR_g // For Case 1, s_lo=s_lo+g
1442 fadd.s1 FR_f_lo = FR_f_lo,FR_A // For Case 2, f_lo=f_lo+A
1447 // a := (B - A) + b Exact.
1448 // Note that a is either 0 or 2^(-128).
1450 // f_lo := (f - f_hi) + A
1451 // f_lo=f-f_hi is exact because either |f| >= |A|, in which
1452 // case f-f_hi is clearly exact; or otherwise, 0<|f|<|A|
1453 // means msb(f) <= msb(A) = 2^(-64) => |f| = 2^(-64).
1454 // If f = 2^(-64), f-f_hi involves cancellation and is
1455 // exact. If f = -2^(-64), then A + f is exact. Hence
1456 // f-f_hi is -A exactly, giving f_lo = 0.
1457 // f_lo := f_lo + a;
1459 // If |f| >= 2^(-33)
1462 // g := A_hi + B_lo;
1464 // s_lo := (f - s_hi) + g;
1468 // A := fadd.fpsr3( A_hi, B_lo )
1469 // B := max( A_hi, B_lo )
1470 // b := min( A_hi, B_lo )
1474 (p10) fcmp.ge.unc.s1 p14, p11 = FR_f_abs,FR_TWOM50
1479 (p13) fms.s1 FR_r_lo = FR_s_hi,FR_D_hi,FR_r_hi //For Case 1, r_lo=s_hi*D_hi+r_hi
1484 // If |f| >= 2^(-50) then
1488 // f_lo := (f_lo + A_lo) + x*p_4
1489 // s_hi := f_hi + f_lo
1490 // s_lo := (f_hi - s_hi) + f_lo
1494 (p14) mov FR_s_hi = FR_f_hi
1499 (p10) fadd.s1 FR_f_lo = FR_f_lo,FR_a
1506 (p14) mov FR_s_lo = FR_f_lo
1511 (p11) fadd.s1 FR_f_lo = FR_f_lo,FR_A_lo
1518 (p11) fma.s1 FR_f_lo = FR_X,FR_p_4,FR_f_lo
1525 (p13) fma.s1 FR_r_lo = FR_s_hi,FR_D_lo,FR_r_lo //For Case 1, r_lo=s_hi*D_lo+r_lo
1530 (p11) fadd.s1 FR_s_hi = FR_f_hi,FR_f_lo
1535 // r_hi := s_hi*D_hi
1536 // r_lo := s_hi*D_hi - r_hi with fma
1537 // r_lo := (s_hi*D_lo + r_lo) + s_lo*D_hi
1540 (p10) fmpy.s1 FR_r_hi = FR_s_hi,FR_D_hi
1545 (p11) fsub.s1 FR_s_lo = FR_f_hi,FR_s_hi
1552 (p10) fms.s1 FR_r_lo = FR_s_hi,FR_D_hi,FR_r_hi
1557 (p11) fadd.s1 FR_s_lo = FR_s_lo,FR_f_lo
1564 (p10) fma.s1 FR_r_lo = FR_s_hi,FR_D_lo,FR_r_lo
1569 // Return N, r_hi, r_lo
1570 // We do not return CASE
1573 fma.s1 FR_r_lo = FR_s_lo,FR_D_hi,FR_r_lo
1578 .endp __libm_pi_by_2_reduce#