4 // Copyright (c) 2000 - 2003, Intel Corporation
5 // All rights reserved.
8 // Redistribution and use in source and binary forms, with or without
9 // modification, are permitted provided that the following conditions are
12 // * Redistributions of source code must retain the above copyright
13 // notice, this list of conditions and the following disclaimer.
15 // * Redistributions in binary form must reproduce the above copyright
16 // notice, this list of conditions and the following disclaimer in the
17 // documentation and/or other materials provided with the distribution.
19 // * The name of Intel Corporation may not be used to endorse or promote
20 // products derived from this software without specific prior written
23 // THIS SOFTWARE IS PROVIDED BY THE COPYRIGHT HOLDERS AND CONTRIBUTORS
24 // "AS IS" AND ANY EXPRESS OR IMPLIED WARRANTIES, INCLUDING, BUT NOT
25 // LIMITED TO, THE IMPLIED WARRANTIES OF MERCHANTABILITY AND FITNESS FOR
26 // A PARTICULAR PURPOSE ARE DISCLAIMED. IN NO EVENT SHALL INTEL OR ITS
27 // CONTRIBUTORS BE LIABLE FOR ANY DIRECT, INDIRECT, INCIDENTAL, SPECIAL,
28 // EXEMPLARY, OR CONSEQUENTIAL DAMAGES (INCLUDING, BUT NOT LIMITED TO,
29 // PROCUREMENT OF SUBSTITUTE GOODS OR SERVICES; LOSS OF USE, DATA, OR
30 // PROFITS; OR BUSINESS INTERRUPTION) HOWEVER CAUSED AND ON ANY THEORY
31 // OF LIABILITY, WHETHER IN CONTRACT, STRICT LIABILITY OR TORT (INCLUDING
32 // NEGLIGENCE OR OTHERWISE) ARISING IN ANY WAY OUT OF THE USE OF THIS
33 // SOFTWARE, EVEN IF ADVISED OF THE POSSIBILITY OF SUCH DAMAGE.
35 // Intel Corporation is the author of this code, and requests that all
36 // problem reports or change requests be submitted to it directly at
37 // http://www.intel.com/software/products/opensource/libraries/num.htm.
40 // 02/02/00 Initial Version
41 // 05/13/02 Rescheduled for speed, changed interface to pass
42 // parameters in fp registers
43 // 02/10/03 Reordered header: .section, .global, .proc, .align;
44 // used data8 for long double data storage
46 //*********************************************************************
47 //*********************************************************************
49 // Function: __libm_pi_by_two_reduce(x) return r, c, and N where
50 // x = N * pi/4 + (r+c) , where |r+c| <= pi/4.
51 // This function is not designed to be used by the
54 //*********************************************************************
56 // Accuracy: Returns double-precision values
58 //*********************************************************************
62 // Floating-Point Registers:
63 // f8 = Input x, return value r
64 // f9 = return value c
67 // General Purpose Registers:
68 // r8 = return value N
71 // Predicate Registers: p6-p14
73 //*********************************************************************
75 // IEEE Special Conditions:
77 // No conditions should be raised.
79 //*********************************************************************
84 // For the forward trigonometric functions sin, cos, sincos, and
85 // tan, the original algorithms for IA 64 handle arguments up to
86 // 1 ulp less than 2^63 in magnitude. For double-extended arguments x,
87 // |x| >= 2^63, this routine returns N and r_hi, r_lo where
89 // x is accurately approximated by
90 // 2*K*pi + N * pi/2 + r_hi + r_lo, |r_hi+r_lo| <= pi/4.
92 // CASE is 1 unless |r_hi + r_lo| < 2^(-33).
94 // The exact value of K is not determined, but that information is
95 // not required in trigonometric function computations.
97 // We first assume the argument x in question satisfies x >= 2^(63).
98 // In particular, it is positive. Negative x can be handled by symmetry:
100 // -x is accurately approximated by
101 // -2*K*pi + (-N) * pi/2 - (r_hi + r_lo), |r_hi+r_lo| <= pi/4.
103 // The idea of the reduction is that
105 // x * 2/pi = N_big + N + f, |f| <= 1/2
107 // Moreover, for double extended x, |f| >= 2^(-75). (This is an
108 // non-obvious fact found by enumeration using a special algorithm
109 // involving continued fraction.) The algorithm described below
110 // calculates N and an accurate approximation of f.
112 // Roughly speaking, an appropriate 256-bit (4 X 64) portion of
113 // 2/pi is multiplied with x to give the desired information.
115 // II. Representation of 2/PI
116 // ==========================
118 // The value of 2/pi in binary fixed-point is
120 // .101000101111100110......
122 // We store 2/pi in a table, starting at the position corresponding
123 // to bit position 63
125 // bit position 63 62 ... 0 -1 -2 -3 -4 -5 -6 -7 .... -16576
127 // 0 0 ... 0 . 1 0 1 0 1 0 1 .... X
130 // |__ implied binary pt
135 // This describes the algorithm in the most natural way using
136 // unsigned integer multiplication. The implementation section
137 // describes how the integer arithmetic is simulated.
139 // STEP 0. Initialization
140 // ----------------------
142 // Let the input argument x be
144 // x = 2^m * ( 1. b_1 b_2 b_3 ... b_63 ), 63 <= m <= 16383.
146 // The first crucial step is to fetch four 64-bit portions of 2/pi.
147 // To fulfill this goal, we calculate the bit position L of the
148 // beginning of these 256-bit quantity by
152 // Note that -16321 <= L <= -1 because 63 <= m <= 16383; and that
153 // the storage of 2/pi is adequate.
155 // Fetch P_1, P_2, P_3, P_4 beginning at bit position L thus:
157 // bit position L L-1 L-2 ... L-63
161 // each b can be 0 or 1. Also, let P_0 be the two bits corresponding to
162 // bit positions L+2 and L+1. So, when each of the P_j is interpreted
163 // with appropriate scaling, we have
165 // 2/pi = P_big + P_0 + (P_1 + P_2 + P_3 + P_4) + P_small
167 // Note that P_big and P_small can be ignored. The reasons are as follow.
168 // First, consider P_big. If P_big = 0, we can certainly ignore it.
169 // Otherwise, P_big >= 2^(L+3). Now,
171 // P_big * ulp(x) >= 2^(L+3) * 2^(m-63)
172 // >= 2^(65-m + m-63 )
175 // Thus, P_big * x is an integer of the form 4*K. So
177 // x = 4*K * (pi/2) + x*(P_0 + P_1 + P_2 + P_3 + P_4)*(pi/2)
178 // + x*P_small*(pi/2).
180 // Hence, P_big*x corresponds to information that can be ignored for
181 // trigonometic function evaluation.
183 // Next, we must estimate the effect of ignoring P_small. The absolute
184 // error made by ignoring P_small is bounded by
186 // |P_small * x| <= ulp(P_4) * x
187 // <= 2^(L-255) * 2^(m+1)
188 // <= 2^(62-m-255 + m + 1)
191 // Since for double-extended precision, x * 2/pi = integer + f,
192 // 0.5 >= |f| >= 2^(-75), the relative error introduced by ignoring
193 // P_small is bounded by 2^(-192+75) <= 2^(-117), which is acceptable.
195 // Further note that if x is split into x_hi + x_lo where x_lo is the
196 // two bits corresponding to bit positions 2^(m-62) and 2^(m-63); then
200 // is also an integer of the form 4*K; and thus can also be ignored.
201 // Let M := P_0 * x_lo which is a small integer. The main part of the
202 // calculation is really the multiplication of x with the four pieces
203 // P_1, P_2, P_3, and P_4.
205 // Unless the reduced argument is extremely small in magnitude, it
206 // suffices to carry out the multiplication of x with P_1, P_2, and
207 // P_3. x*P_4 will be carried out and added on as a correction only
208 // when it is found to be needed. Note also that x*P_4 need not be
209 // computed exactly. A straightforward multiplication suffices since
210 // the rounding error thus produced would be bounded by 2^(-3*64),
211 // that is 2^(-192) which is small enough as the reduced argument
212 // is bounded from below by 2^(-75).
214 // Now that we have four 64-bit data representing 2/pi and a
215 // 64-bit x. We first need to calculate a highly accurate product
216 // of x and P_1, P_2, P_3. This is best understood as integer
220 // STEP 1. Multiplication
221 // ----------------------
224 // --------- --------- ---------
225 // | P_1 | | P_2 | | P_3 |
226 // --------- --------- ---------
231 // ----------------------------------------------------
233 // --------- ---------
235 // --------- ---------
238 // --------- ---------
240 // --------- ---------
243 // --------- ---------
245 // --------- ---------
247 // ====================================================
248 // --------- --------- --------- ---------
249 // | S_0 | | S_1 | | S_2 | | S_3 |
250 // --------- --------- --------- ---------
254 // STEP 2. Get N and f
255 // -------------------
257 // Conceptually, after the individual pieces S_0, S_1, ..., are obtained,
258 // we have to sum them and obtain an integer part, N, and a fraction, f.
259 // Here, |f| <= 1/2, and N is an integer. Note also that N need only to
260 // be known to module 2^k, k >= 2. In the case when |f| is small enough,
261 // we would need to add in the value x*P_4.
264 // STEP 3. Get reduced argument
265 // ----------------------------
267 // The value f is not yet the reduced argument that we seek. The
270 // x * 2/pi = 4K + N + f
274 // x = 2*K*pi + N * pi/2 + f * (pi/2).
276 // Thus, the reduced argument is given by
278 // reduced argument = f * pi/2.
280 // This multiplication must be performed to extra precision.
282 // IV. Implementation
283 // ==================
285 // Step 0. Initialization
286 // ----------------------
288 // Set sgn_x := sign(x); x := |x|; x_lo := 2 lsb of x.
290 // In memory, 2/pi is stored contiguously as
292 // 0x00000000 0x00000000 0xA2F....
294 // |__ implied binary bit
296 // Given x = 2^m * 1.xxxx...xxx; we calculate L := 62 - m. Thus
297 // -1 <= L <= -16321. We fetch from memory 5 integer pieces of data.
299 // P_0 is the two bits corresponding to bit positions L+2 and L+1
300 // P_1 is the 64-bit starting at bit position L
301 // P_2 is the 64-bit starting at bit position L-64
302 // P_3 is the 64-bit starting at bit position L-128
303 // P_4 is the 64-bit starting at bit position L-192
305 // For example, if m = 63, P_0 would be 0 and P_1 would look like
308 // If m = 65, P_0 would be the two msb of 0xA, thus, P_0 is 10 in binary.
309 // P_1 in binary would be 1 0 0 0 1 0 1 1 1 1 ....
311 // Step 1. Multiplication
312 // ----------------------
314 // At this point, P_1, P_2, P_3, P_4 are integers. They are
315 // supposed to be interpreted as
318 // 2^(L-63-64) * P_2;
319 // 2^(L-63-128) * P_3;
320 // 2^(L-63-192) * P_4;
322 // Since each of them need to be multiplied to x, we would scale
323 // both x and the P_j's by some convenient factors: scale each
324 // of P_j's up by 2^(63-L), and scale x down by 2^(L-63).
326 // p_1 := fcvt.xf ( P_1 )
327 // p_2 := fcvt.xf ( P_2 ) * 2^(-64)
328 // p_3 := fcvt.xf ( P_3 ) * 2^(-128)
329 // p_4 := fcvt.xf ( P_4 ) * 2^(-192)
330 // x := replace exponent of x by -1
331 // because 2^m * 1.xxxx...xxx * 2^(L-63)
332 // is 2^(-1) * 1.xxxx...xxx
334 // We are now faced with the task of computing the following
336 // --------- --------- ---------
337 // | P_1 | | P_2 | | P_3 |
338 // --------- --------- ---------
343 // ----------------------------------------------------
345 // --------- ---------
347 // --------- ---------
349 // --------- ---------
351 // --------- ---------
353 // --------- ---------
355 // --------- ---------
357 // ====================================================
358 // ----------- --------- --------- ---------
359 // | S_0 | | S_1 | | S_2 | | S_3 |
360 // ----------- --------- --------- ---------
362 // | |___ binary point
364 // |___ possibly one more bit
366 // Let FPSR3 be set to round towards zero with widest precision
367 // and exponent range. Unless an explicit FPSR is given,
368 // round-to-nearest with widest precision and exponent range is
371 // Define sigma_C := 2^63; sigma_B := 2^(-1); sigma_C := 2^(-65).
373 // Tmp_C := fmpy.fpsr3( x, p_1 );
374 // If Tmp_C >= sigma_C then
376 // C_lo := x*p_1 - C_hi ...fma, exact
378 // C_hi := fadd.fpsr3(sigma_C, Tmp_C) - sigma_C
379 // ...subtraction is exact, regardless
380 // ...of rounding direction
381 // C_lo := x*p_1 - C_hi ...fma, exact
384 // Tmp_B := fmpy.fpsr3( x, p_2 );
385 // If Tmp_B >= sigma_B then
387 // B_lo := x*p_2 - B_hi ...fma, exact
389 // B_hi := fadd.fpsr3(sigma_B, Tmp_B) - sigma_B
390 // ...subtraction is exact, regardless
391 // ...of rounding direction
392 // B_lo := x*p_2 - B_hi ...fma, exact
395 // Tmp_A := fmpy.fpsr3( x, p_3 );
396 // If Tmp_A >= sigma_A then
398 // A_lo := x*p_3 - A_hi ...fma, exact
400 // A_hi := fadd.fpsr3(sigma_A, Tmp_A) - sigma_A
401 // ...subtraction is exact, regardless
402 // ...of rounding direction
403 // A_lo := x*p_3 - A_hi ...fma, exact
406 // ...Note that C_hi is of integer value. We need only the
407 // ...last few bits. Thus we can ensure C_hi is never a big
408 // ...integer, freeing us from overflow worry.
410 // Tmp_C := fadd.fpsr3( C_hi, 2^(70) ) - 2^(70);
411 // ...Tmp_C is the upper portion of C_hi
412 // C_hi := C_hi - Tmp_C
413 // ...0 <= C_hi < 2^7
415 // Step 2. Get N and f
416 // -------------------
418 // At this point, we have all the components to obtain
419 // S_0, S_1, S_2, S_3 and thus N and f. We start by adding
420 // C_lo and B_hi. This sum together with C_hi gives a good
421 // estimation of N and f.
423 // A := fadd.fpsr3( B_hi, C_lo )
424 // B := max( B_hi, C_lo )
425 // b := min( B_hi, C_lo )
427 // a := (B - A) + b ...exact. Note that a is either 0
430 // N := round_to_nearest_integer_value( A );
431 // f := A - N; ...exact because lsb(A) >= 2^(-64)
432 // ...and |f| <= 1/2.
434 // f := f + a ...exact because a is 0 or 2^(-64);
435 // ...the msb of the sum is <= 1/2
436 // ...lsb >= 2^(-64).
438 // N := convert to integer format( C_hi + N );
442 // If sgn_x == 1 (that is original x was negative)
444 // ...this maintains N to be non-negative, but still
445 // ...equivalent to the (negated N) mod 4.
454 // s_lo := (f - s_hi) + g;
460 // A := fadd.fpsr3( A_hi, B_lo )
461 // B := max( A_hi, B_lo )
462 // b := min( A_hi, B_lo )
464 // a := (B - A) + b ...exact. Note that a is either 0
468 // f_lo := (f - f_hi) + A;
470 // ...f-f_hi is exact because either |f| >= |A|, in which
471 // ...case f-f_hi is clearly exact; or otherwise, 0<|f|<|A|
472 // ...means msb(f) <= msb(A) = 2^(-64) => |f| = 2^(-64).
473 // ...If f = 2^(-64), f-f_hi involves cancellation and is
474 // ...exact. If f = -2^(-64), then A + f is exact. Hence
475 // ...f-f_hi is -A exactly, giving f_lo = 0.
479 // If |f| >= 2^(-50) then
483 // f_lo := (f_lo + A_lo) + x*p_4
484 // s_hi := f_hi + f_lo
485 // s_lo := (f_hi - s_hi) + f_lo
490 // Step 3. Get reduced argument
491 // ----------------------------
493 // If sgn_x == 0 (that is original x is positive)
495 // D_hi := Pi_by_2_hi
496 // D_lo := Pi_by_2_lo
497 // ...load from table
501 // D_hi := neg_Pi_by_2_hi
502 // D_lo := neg_Pi_by_2_lo
503 // ...load from table
507 // r_lo := s_hi*D_hi - r_hi ...fma
508 // r_lo := (s_hi*D_lo + r_lo) + s_lo*D_hi
510 // Return N, r_hi, r_lo
591 LOCAL_OBJECT_START(Constants_Bits_of_2_by_pi)
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702 data8 0xC68351103ED84871,0xDDDD1C2DA118AF46
703 data8 0x2C21D7F359987AD9,0xC0549EFA864FFC06
704 data8 0x56AE79E536228922,0xAD38DC9367AAE855
705 data8 0x3826829BE7CAA40D,0x51B133990ED7A948
706 data8 0x0569F0B265A7887F,0x974C8836D1F9B392
707 data8 0x214A827B21CF98DC,0x9F405547DC3A74E1
708 data8 0x42EB67DF9DFE5FD4,0x5EA4677B7AACBAA2
709 data8 0xF65523882B55BA41,0x086E59862A218347
710 data8 0x39E6E389D49EE540,0xFB49E956FFCA0F1C
711 data8 0x8A59C52BFA94C5C1,0xD3CFC50FAE5ADB86
712 data8 0xC5476243853B8621,0x94792C8761107B4C
713 data8 0x2A1A2C8012BF4390,0x2688893C78E4C4A8
714 data8 0x7BDBE5C23AC4EAF4,0x268A67F7BF920D2B
715 data8 0xA365B1933D0B7CBD,0xDC51A463DD27DDE1
716 data8 0x6919949A9529A828,0xCE68B4ED09209F44
717 data8 0xCA984E638270237C,0x7E32B90F8EF5A7E7
718 data8 0x561408F1212A9DB5,0x4D7E6F5119A5ABF9
719 data8 0xB5D6DF8261DD9602,0x36169F3AC4A1A283
720 data8 0x6DED727A8D39A9B8,0x825C326B5B2746ED
721 data8 0x34007700D255F4FC,0x4D59018071E0E13F
722 data8 0x89B295F364A8F1AE,0xA74B38FC4CEAB2BB
723 LOCAL_OBJECT_END(Constants_Bits_of_2_by_pi)
725 LOCAL_OBJECT_START(Constants_Bits_of_pi_by_2)
726 data8 0xC90FDAA22168C234,0x00003FFF
727 data8 0xC4C6628B80DC1CD1,0x00003FBF
728 LOCAL_OBJECT_END(Constants_Bits_of_pi_by_2)
731 .global __libm_pi_by_2_reduce#
732 .proc __libm_pi_by_2_reduce#
735 __libm_pi_by_2_reduce:
738 // Place the two-piece result r (r_hi) in f8 and c (r_lo) in f9
739 // N is returned in r8
742 alloc r34 = ar.pfs,2,34,0,0
743 fsetc.s3 0x00,0x7F // Set sf3 to round to zero, 82-bit prec, td, ftz
747 addl GR_BASE = @ltoff(Constants_Bits_of_2_by_pi#), gp
749 mov GR_BIASL63 = 0x1003E
755 // 0 0 0 0 0. 1 0 1 0
756 // M 0 1 2 .... 63, 64 65 ... 127, 128
757 // ---------------------------------------------
758 // Segment 0. 1 , 2 , 3
759 // START = M - 63 M = 128 becomes 65
760 // LENGTH1 = START & 0x3F 65 become position 1
761 // SEGMENT = shr(START,6) + 1 0 maps to 1, 64 maps to 2,
762 // LENGTH2 = 64 - LENGTH1
763 // Address_BASE = shladd(SEGMENT,3) + BASE
767 getf.exp GR_Exp_x = FR_input_X
768 ld8 GR_BASE = [GR_BASE]
769 mov GR_TEMP5 = 0x0FFFE
773 // Define sigma_C := 2^63; sigma_B := 2^(-1); sigma_A := 2^(-65).
775 getf.sig GR_x_lo = FR_input_X
776 mov GR_TEMP6 = 0x0FFBE
781 // Special Code for testing DE arguments
782 // movl GR_BIASL63 = 0x0000000000013FFE
783 // movl GR_x_lo = 0xFFFFFFFFFFFFFFFF
784 // setf.exp FR_X = GR_BIASL63
785 // setf.sig FR_ScaleP3 = GR_x_lo
786 // fmerge.se FR_X = FR_X,FR_ScaleP3
787 // Set sgn_x := sign(x); x := |x|; x_lo := 2 lsb of x.
788 // 2/pi is stored contiguously as
789 // 0x00000000 0x00000000.0xA2F....
790 // M = EXP - BIAS ( M >= 63)
791 // Given x = 2^m * 1.xxxx...xxx; we calculate L := 62 - m.
792 // Thus -1 <= L <= -16321.
794 setf.exp FR_sigma_B = GR_TEMP5
795 setf.exp FR_sigma_A = GR_TEMP6
796 extr.u GR_M = GR_Exp_x,0,17
801 and GR_x_lo = 0x03,GR_x_lo
802 sub GR_START = GR_M,GR_BIASL63
803 add GR_BASE = 8,GR_BASE // To effectively add 1 to SEGMENT
808 and GR_LENGTH1 = 0x3F,GR_START
809 shr.u GR_SEGMENT = GR_START,6
815 shladd GR_BASE = GR_SEGMENT,3,GR_BASE
816 sub GR_LENGTH2 = 0x40,GR_LENGTH1
817 cmp.le p6,p7 = 0x2,GR_LENGTH1
821 // P_0 is the two bits corresponding to bit positions L+2 and L+1
822 // P_1 is the 64-bit starting at bit position L
823 // P_2 is the 64-bit starting at bit position L-64
824 // P_3 is the 64-bit starting at bit position L-128
825 // P_4 is the 64-bit starting at bit position L-192
826 // P_1 is made up of Alo and Bhi
827 // P_1 = deposit Alo, position 0, length2 into P_1,position length1
828 // deposit Bhi, position length2, length1 into P_1, position 0
829 // P_2 is made up of Blo and Chi
830 // P_2 = deposit Blo, position 0, length2 into P_2, position length1
831 // deposit Chi, position length2, length1 into P_2, position 0
832 // P_3 is made up of Clo and Dhi
833 // P_3 = deposit Clo, position 0, length2 into P_3, position length1
834 // deposit Dhi, position length2, length1 into P_3, position 0
835 // P_4 is made up of Clo and Dhi
836 // P_4 = deposit Dlo, position 0, length2 into P_4, position length1
837 // deposit Ehi, position length2, length1 into P_4, position 0
839 ld8 GR_A = [GR_BASE],8
840 fabs FR_X = FR_input_X
841 (p7) cmp.eq.unc p8,p9 = 0x1,GR_LENGTH1
845 // ld_64 A at Base and increment Base by 8
846 // ld_64 B at Base and increment Base by 8
847 // ld_64 C at Base and increment Base by 8
848 // ld_64 D at Base and increment Base by 8
849 // ld_64 E at Base and increment Base by 8
851 // ---------------------
852 // A, B, C, D, and E look like | length1 | length2 |
853 // ---------------------
856 ld8 GR_B = [GR_BASE],8
857 movl GR_rshf = 0x43e8000000000000 // 1.10000 2^63 for right shift N_fix
862 ld8 GR_C = [GR_BASE],8
864 (p8) extr.u GR_Temp = GR_A,63,1
869 // P_0 = deposit Ahi, position length2, 2 bit into P_0 at position 0.
871 ld8 GR_D = [GR_BASE],8
872 shl GR_TEMP1 = GR_A,GR_LENGTH1 // MM instruction
873 (p6) shr.u GR_P_0 = GR_A,GR_LENGTH2 // MM instruction
878 ld8 GR_E = [GR_BASE],-40
879 shl GR_TEMP2 = GR_B,GR_LENGTH1 // MM instruction
880 shr.u GR_P_1 = GR_B,GR_LENGTH2 // MM instruction
885 // Load 16 bit of ASUB from (Base_Address_of_A - 2)
890 // Deposit element 63 from Ahi and place in element 0 of P_0.
895 (p7) ld2 GR_ASUB = [GR_BASE],8
896 shl GR_TEMP3 = GR_C,GR_LENGTH1 // MM instruction
897 shr.u GR_P_2 = GR_C,GR_LENGTH2 // MM instruction
902 setf.d FR_RSHF = GR_rshf // Form right shift const 1.100 * 2^63
903 shl GR_TEMP4 = GR_D,GR_LENGTH1 // MM instruction
904 shr.u GR_P_3 = GR_D,GR_LENGTH2 // MM instruction
909 (p7) and GR_P_0 = 0x03,GR_ASUB
910 (p6) and GR_P_0 = 0x03,GR_P_0
911 shr.u GR_P_4 = GR_E,GR_LENGTH2 // MM instruction
917 or GR_P_1 = GR_P_1,GR_TEMP1
918 (p8) and GR_P_0 = 0x1,GR_P_0
923 setf.sig FR_p_1 = GR_P_1
924 or GR_P_2 = GR_P_2,GR_TEMP2
925 (p8) shladd GR_P_0 = GR_P_0,1,GR_Temp
930 setf.sig FR_p_2 = GR_P_2
931 or GR_P_3 = GR_P_3,GR_TEMP3
932 fmerge.se FR_X = FR_sigma_B,FR_X
937 setf.sig FR_p_3 = GR_P_3
938 or GR_P_4 = GR_P_4,GR_TEMP4
939 pmpy2.r GR_M = GR_P_0,GR_x_lo
943 // P_1, P_2, P_3, P_4 are integers. They should be
945 // 2^(L-63-64) * P_2;
946 // 2^(L-63-128) * P_3;
947 // 2^(L-63-192) * P_4;
948 // Since each of them need to be multiplied to x, we would scale
949 // both x and the P_j's by some convenient factors: scale each
950 // of P_j's up by 2^(63-L), and scale x down by 2^(L-63).
951 // p_1 := fcvt.xf ( P_1 )
952 // p_2 := fcvt.xf ( P_2 ) * 2^(-64)
953 // p_3 := fcvt.xf ( P_3 ) * 2^(-128)
954 // p_4 := fcvt.xf ( P_4 ) * 2^(-192)
955 // x= Set x's exp to -1 because 2^m*1.x...x *2^(L-63)=2^(-1)*1.x...xxx
956 // --------- --------- ---------
957 // | P_1 | | P_2 | | P_3 |
958 // --------- --------- ---------
962 // ----------------------------------------------------
963 // --------- ---------
965 // --------- ---------
966 // --------- ---------
968 // --------- ---------
969 // --------- ---------
971 // --------- ---------
972 // ====================================================
973 // ----------- --------- --------- ---------
974 // | S_0 | | S_1 | | S_2 | | S_3 |
975 // ----------- --------- --------- ---------
976 // | |___ binary point
977 // |___ possibly one more bit
979 // Let FPSR3 be set to round towards zero with widest precision
980 // and exponent range. Unless an explicit FPSR is given,
981 // round-to-nearest with widest precision and exponent range is
984 setf.sig FR_p_4 = GR_P_4
985 mov GR_TEMP1 = 0x0FFBF
991 setf.exp FR_ScaleP2 = GR_TEMP1
992 mov GR_TEMP2 = 0x0FF7F
998 setf.exp FR_ScaleP3 = GR_TEMP2
999 mov GR_TEMP4 = 0x1003E
1005 setf.exp FR_sigma_C = GR_TEMP4
1006 mov GR_Temp = 0x0FFDE
1007 fcvt.xuf.s1 FR_p_1 = FR_p_1
1012 setf.exp FR_TWOM33 = GR_Temp
1013 fcvt.xuf.s1 FR_p_2 = FR_p_2
1020 fcvt.xuf.s1 FR_p_3 = FR_p_3
1027 fcvt.xuf.s1 FR_p_4 = FR_p_4
1032 // Tmp_C := fmpy.fpsr3( x, p_1 );
1033 // Tmp_B := fmpy.fpsr3( x, p_2 );
1034 // Tmp_A := fmpy.fpsr3( x, p_3 );
1035 // If Tmp_C >= sigma_C then
1037 // C_lo := x*p_1 - C_hi ...fma, exact
1039 // C_hi := fadd.fpsr3(sigma_C, Tmp_C) - sigma_C
1040 // C_lo := x*p_1 - C_hi ...fma, exact
1042 // If Tmp_B >= sigma_B then
1044 // B_lo := x*p_2 - B_hi ...fma, exact
1046 // B_hi := fadd.fpsr3(sigma_B, Tmp_B) - sigma_B
1047 // B_lo := x*p_2 - B_hi ...fma, exact
1049 // If Tmp_A >= sigma_A then
1051 // A_lo := x*p_3 - A_hi ...fma, exact
1053 // A_hi := fadd.fpsr3(sigma_A, Tmp_A) - sigma_A
1054 // Exact, regardless ...of rounding direction
1055 // A_lo := x*p_3 - A_hi ...fma, exact
1059 fmpy.s3 FR_Tmp_C = FR_X,FR_p_1
1065 mov GR_TEMP3 = 0x0FF3F
1066 fmpy.s1 FR_p_2 = FR_p_2,FR_ScaleP2
1072 setf.exp FR_ScaleP4 = GR_TEMP3
1073 mov GR_TEMP4 = 0x10045
1074 fmpy.s1 FR_p_3 = FR_p_3,FR_ScaleP3
1080 fadd.s3 FR_C_hi = FR_sigma_C,FR_Tmp_C // For Tmp_C < sigma_C case
1086 setf.exp FR_Tmp2_C = GR_TEMP4
1088 fmpy.s3 FR_Tmp_B = FR_X,FR_p_2
1093 addl GR_BASE = @ltoff(Constants_Bits_of_pi_by_2#), gp
1094 fcmp.ge.s1 p12, p9 = FR_Tmp_C,FR_sigma_C
1099 fmpy.s3 FR_Tmp_A = FR_X,FR_p_3
1105 ld8 GR_BASE = [GR_BASE]
1106 (p12) mov FR_C_hi = FR_Tmp_C
1111 (p9) fsub.s1 FR_C_hi = FR_C_hi,FR_sigma_C
1119 // Step 3. Get reduced argument
1120 // If sgn_x == 0 (that is original x is positive)
1121 // D_hi := Pi_by_2_hi
1122 // D_lo := Pi_by_2_lo
1125 // D_hi := neg_Pi_by_2_hi
1126 // D_lo := neg_Pi_by_2_lo
1132 fmpy.s1 FR_p_4 = FR_p_4,FR_ScaleP4
1137 fadd.s3 FR_B_hi = FR_sigma_B,FR_Tmp_B // For Tmp_B < sigma_B case
1144 fadd.s3 FR_A_hi = FR_sigma_A,FR_Tmp_A // For Tmp_A < sigma_A case
1151 fcmp.ge.s1 p13, p10 = FR_Tmp_B,FR_sigma_B
1156 fms.s1 FR_C_lo = FR_X,FR_p_1,FR_C_hi
1162 ldfe FR_D_hi = [GR_BASE],16
1163 fcmp.ge.s1 p14, p11 = FR_Tmp_A,FR_sigma_A
1169 ldfe FR_D_lo = [GR_BASE]
1170 (p13) mov FR_B_hi = FR_Tmp_B
1175 (p10) fsub.s1 FR_B_hi = FR_B_hi,FR_sigma_B
1182 (p14) mov FR_A_hi = FR_Tmp_A
1187 (p11) fsub.s1 FR_A_hi = FR_A_hi,FR_sigma_A
1192 // Note that C_hi is of integer value. We need only the
1193 // last few bits. Thus we can ensure C_hi is never a big
1194 // integer, freeing us from overflow worry.
1195 // Tmp_C := fadd.fpsr3( C_hi, 2^(70) ) - 2^(70);
1196 // Tmp_C is the upper portion of C_hi
1199 fadd.s3 FR_Tmp_C = FR_C_hi,FR_Tmp2_C
1200 tbit.z p12,p9 = GR_Exp_x, 17
1206 fms.s1 FR_B_lo = FR_X,FR_p_2,FR_B_hi
1211 fadd.s3 FR_A = FR_B_hi,FR_C_lo
1218 fms.s1 FR_A_lo = FR_X,FR_p_3,FR_A_hi
1225 fsub.s1 FR_Tmp_C = FR_Tmp_C,FR_Tmp2_C
1230 // *******************
1231 // Step 2. Get N and f
1232 // *******************
1233 // We have all the components to obtain
1234 // S_0, S_1, S_2, S_3 and thus N and f. We start by adding
1235 // C_lo and B_hi. This sum together with C_hi estimates
1237 // A := fadd.fpsr3( B_hi, C_lo )
1238 // B := max( B_hi, C_lo )
1239 // b := min( B_hi, C_lo )
1242 fmax.s1 FR_B = FR_B_hi,FR_C_lo
1247 // We use a right-shift trick to get the integer part of A into the rightmost
1248 // bits of the significand by adding 1.1000..00 * 2^63. This operation is good
1249 // if |A| < 2^61, which it is in this case. We are doing this to save a few
1250 // cycles over using fcvt.fx followed by fnorm. The second step of the trick
1251 // is to subtract the same constant to float the rounded integer into a fp reg.
1255 // N := round_to_nearest_integer_value( A );
1256 fma.s1 FR_N_fix = FR_A, f1, FR_RSHF
1263 fmin.s1 FR_b = FR_B_hi,FR_C_lo
1268 // C_hi := C_hi - Tmp_C ...0 <= C_hi < 2^7
1269 fsub.s1 FR_C_hi = FR_C_hi,FR_Tmp_C
1276 // a := (B - A) + b: Exact - note that a is either 0 or 2^(-64).
1277 fsub.s1 FR_a = FR_B,FR_A
1284 fms.s1 FR_N = FR_N_fix, f1, FR_RSHF
1291 fadd.s1 FR_a = FR_a,FR_b
1296 // f := A - N; Exact because lsb(A) >= 2^(-64) and |f| <= 1/2.
1297 // N := convert to integer format( C_hi + N );
1302 fsub.s1 FR_f = FR_A,FR_N
1307 fadd.s1 FR_N = FR_N,FR_C_hi
1314 (p9) fsub.s1 FR_D_hi = f0, FR_D_hi
1319 (p9) fsub.s1 FR_D_lo = f0, FR_D_lo
1326 fadd.s1 FR_g = FR_A_hi,FR_B_lo // For Case 1, g=A_hi+B_lo
1331 fadd.s3 FR_A = FR_A_hi,FR_B_lo // For Case 2, A=A_hi+B_lo w/ sf3
1337 mov GR_Temp = 0x0FFCD // For Case 2, exponent of 2^-50
1338 fmax.s1 FR_B = FR_A_hi,FR_B_lo // For Case 2, B=max(A_hi,B_lo)
1343 // f = f + a Exact because a is 0 or 2^(-64);
1344 // the msb of the sum is <= 1/2 and lsb >= 2^(-64).
1346 setf.exp FR_TWOM50 = GR_Temp // For Case 2, form 2^-50
1347 fcvt.fx.s1 FR_N = FR_N
1352 fadd.s1 FR_f = FR_f,FR_a
1359 fmin.s1 FR_b = FR_A_hi,FR_B_lo // For Case 2, b=min(A_hi,B_lo)
1366 fsub.s1 FR_a = FR_B,FR_A // For Case 2, a=B-A
1373 fadd.s1 FR_s_hi = FR_f,FR_g // For Case 1, s_hi=f+g
1378 fadd.s1 FR_f_hi = FR_A,FR_f // For Case 2, f_hi=A+f
1385 fabs FR_f_abs = FR_f
1391 getf.sig GR_N = FR_N
1392 fsetc.s3 0x7F,0x40 // Reset sf3 to user settings + td
1399 fsub.s1 FR_s_lo = FR_f,FR_s_hi // For Case 1, s_lo=f-s_hi
1404 fsub.s1 FR_f_lo = FR_f,FR_f_hi // For Case 2, f_lo=f-f_hi
1411 fmpy.s1 FR_r_hi = FR_s_hi,FR_D_hi // For Case 1, r_hi=s_hi*D_hi
1416 fadd.s1 FR_a = FR_a,FR_b // For Case 2, a=a+b
1422 // If sgn_x == 1 (that is original x was negative)
1424 // this maintains N to be non-negative, but still
1425 // equivalent to the (negated N) mod 4.
1428 add GR_N = GR_N,GR_M
1429 fcmp.ge.s1 p13, p10 = FR_f_abs,FR_TWOM33
1430 mov GR_Temp = 0x00400
1435 (p9) sub GR_N = GR_Temp,GR_N
1436 fadd.s1 FR_s_lo = FR_s_lo,FR_g // For Case 1, s_lo=s_lo+g
1441 fadd.s1 FR_f_lo = FR_f_lo,FR_A // For Case 2, f_lo=f_lo+A
1446 // a := (B - A) + b Exact.
1447 // Note that a is either 0 or 2^(-128).
1449 // f_lo := (f - f_hi) + A
1450 // f_lo=f-f_hi is exact because either |f| >= |A|, in which
1451 // case f-f_hi is clearly exact; or otherwise, 0<|f|<|A|
1452 // means msb(f) <= msb(A) = 2^(-64) => |f| = 2^(-64).
1453 // If f = 2^(-64), f-f_hi involves cancellation and is
1454 // exact. If f = -2^(-64), then A + f is exact. Hence
1455 // f-f_hi is -A exactly, giving f_lo = 0.
1456 // f_lo := f_lo + a;
1458 // If |f| >= 2^(-33)
1461 // g := A_hi + B_lo;
1463 // s_lo := (f - s_hi) + g;
1467 // A := fadd.fpsr3( A_hi, B_lo )
1468 // B := max( A_hi, B_lo )
1469 // b := min( A_hi, B_lo )
1473 (p10) fcmp.ge.unc.s1 p14, p11 = FR_f_abs,FR_TWOM50
1478 (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
1483 // If |f| >= 2^(-50) then
1487 // f_lo := (f_lo + A_lo) + x*p_4
1488 // s_hi := f_hi + f_lo
1489 // s_lo := (f_hi - s_hi) + f_lo
1493 (p14) mov FR_s_hi = FR_f_hi
1498 (p10) fadd.s1 FR_f_lo = FR_f_lo,FR_a
1505 (p14) mov FR_s_lo = FR_f_lo
1510 (p11) fadd.s1 FR_f_lo = FR_f_lo,FR_A_lo
1517 (p11) fma.s1 FR_f_lo = FR_X,FR_p_4,FR_f_lo
1524 (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
1529 (p11) fadd.s1 FR_s_hi = FR_f_hi,FR_f_lo
1534 // r_hi := s_hi*D_hi
1535 // r_lo := s_hi*D_hi - r_hi with fma
1536 // r_lo := (s_hi*D_lo + r_lo) + s_lo*D_hi
1539 (p10) fmpy.s1 FR_r_hi = FR_s_hi,FR_D_hi
1544 (p11) fsub.s1 FR_s_lo = FR_f_hi,FR_s_hi
1551 (p10) fms.s1 FR_r_lo = FR_s_hi,FR_D_hi,FR_r_hi
1556 (p11) fadd.s1 FR_s_lo = FR_s_lo,FR_f_lo
1563 (p10) fma.s1 FR_r_lo = FR_s_hi,FR_D_lo,FR_r_lo
1568 // Return N, r_hi, r_lo
1569 // We do not return CASE
1572 fma.s1 FR_r_lo = FR_s_lo,FR_D_hi,FR_r_lo
1577 .endp __libm_pi_by_2_reduce#