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Diffstat (limited to 'sysdeps/ia64/fpu/libm_reduce.S')
| -rw-r--r-- | sysdeps/ia64/fpu/libm_reduce.S | 1577 |
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diff --git a/sysdeps/ia64/fpu/libm_reduce.S b/sysdeps/ia64/fpu/libm_reduce.S deleted file mode 100644 index 01f16e423c..0000000000 --- a/sysdeps/ia64/fpu/libm_reduce.S +++ /dev/null @@ -1,1577 +0,0 @@ -.file "libm_reduce.s" - - -// Copyright (c) 2000 - 2003, Intel Corporation -// All rights reserved. -// -// -// Redistribution and use in source and binary forms, with or without -// modification, are permitted provided that the following conditions are -// met: -// -// * Redistributions of source code must retain the above copyright -// notice, this list of conditions and the following disclaimer. -// -// * Redistributions in binary form must reproduce the above copyright -// notice, this list of conditions and the following disclaimer in the -// documentation and/or other materials provided with the distribution. -// -// * The name of Intel Corporation may not be used to endorse or promote -// products derived from this software without specific prior written -// permission. - -// THIS SOFTWARE IS PROVIDED BY THE COPYRIGHT HOLDERS AND CONTRIBUTORS -// "AS IS" AND ANY EXPRESS OR IMPLIED WARRANTIES, INCLUDING, BUT NOT -// LIMITED TO, THE IMPLIED WARRANTIES OF MERCHANTABILITY AND FITNESS FOR -// A PARTICULAR PURPOSE ARE DISCLAIMED. IN NO EVENT SHALL INTEL OR ITS -// CONTRIBUTORS BE LIABLE FOR ANY DIRECT, INDIRECT, INCIDENTAL, SPECIAL, -// EXEMPLARY, OR CONSEQUENTIAL DAMAGES (INCLUDING, BUT NOT LIMITED TO, -// PROCUREMENT OF SUBSTITUTE GOODS OR SERVICES; LOSS OF USE, DATA, OR -// PROFITS; OR BUSINESS INTERRUPTION) HOWEVER CAUSED AND ON ANY THEORY -// OF LIABILITY, WHETHER IN CONTRACT, STRICT LIABILITY OR TORT (INCLUDING -// NEGLIGENCE OR OTHERWISE) ARISING IN ANY WAY OUT OF THE USE OF THIS -// SOFTWARE, EVEN IF ADVISED OF THE POSSIBILITY OF SUCH DAMAGE. -// -// Intel Corporation is the author of this code, and requests that all -// problem reports or change requests be submitted to it directly at -// http://www.intel.com/software/products/opensource/libraries/num.htm. -// -// History: -// 02/02/00 Initial Version -// 05/13/02 Rescheduled for speed, changed interface to pass -// parameters in fp registers -// 02/10/03 Reordered header: .section, .global, .proc, .align; -// used data8 for long double data storage -// -//********************************************************************* -//********************************************************************* -// -// Function: __libm_pi_by_two_reduce(x) return r, c, and N where -// x = N * pi/4 + (r+c) , where |r+c| <= pi/4. -// This function is not designed to be used by the -// general user. -// -//********************************************************************* -// -// Accuracy: Returns double-precision values -// -//********************************************************************* -// -// Resources Used: -// -// Floating-Point Registers: -// f8 = Input x, return value r -// f9 = return value c -// f32-f70 -// -// General Purpose Registers: -// r8 = return value N -// r34-r64 -// -// Predicate Registers: p6-p14 -// -//********************************************************************* -// -// IEEE Special Conditions: -// -// No conditions should be raised. -// -//********************************************************************* -// -// I. Introduction -// =============== -// -// For the forward trigonometric functions sin, cos, sincos, and -// tan, the original algorithms for IA 64 handle arguments up to -// 1 ulp less than 2^63 in magnitude. For double-extended arguments x, -// |x| >= 2^63, this routine returns N and r_hi, r_lo where -// -// x is accurately approximated by -// 2*K*pi + N * pi/2 + r_hi + r_lo, |r_hi+r_lo| <= pi/4. -// CASE = 1 or 2. -// CASE is 1 unless |r_hi + r_lo| < 2^(-33). -// -// The exact value of K is not determined, but that information is -// not required in trigonometric function computations. -// -// We first assume the argument x in question satisfies x >= 2^(63). -// In particular, it is positive. Negative x can be handled by symmetry: -// -// -x is accurately approximated by -// -2*K*pi + (-N) * pi/2 - (r_hi + r_lo), |r_hi+r_lo| <= pi/4. -// -// The idea of the reduction is that -// -// x * 2/pi = N_big + N + f, |f| <= 1/2 -// -// Moreover, for double extended x, |f| >= 2^(-75). (This is an -// non-obvious fact found by enumeration using a special algorithm -// involving continued fraction.) The algorithm described below -// calculates N and an accurate approximation of f. -// -// Roughly speaking, an appropriate 256-bit (4 X 64) portion of -// 2/pi is multiplied with x to give the desired information. -// -// II. Representation of 2/PI -// ========================== -// -// The value of 2/pi in binary fixed-point is -// -// .101000101111100110...... -// -// We store 2/pi in a table, starting at the position corresponding -// to bit position 63 -// -// bit position 63 62 ... 0 -1 -2 -3 -4 -5 -6 -7 .... -16576 -// -// 0 0 ... 0 . 1 0 1 0 1 0 1 .... X -// -// ^ -// |__ implied binary pt -// -// III. Algorithm -// ============== -// -// This describes the algorithm in the most natural way using -// unsigned integer multiplication. The implementation section -// describes how the integer arithmetic is simulated. -// -// STEP 0. Initialization -// ---------------------- -// -// Let the input argument x be -// -// x = 2^m * ( 1. b_1 b_2 b_3 ... b_63 ), 63 <= m <= 16383. -// -// The first crucial step is to fetch four 64-bit portions of 2/pi. -// To fulfill this goal, we calculate the bit position L of the -// beginning of these 256-bit quantity by -// -// L := 62 - m. -// -// Note that -16321 <= L <= -1 because 63 <= m <= 16383; and that -// the storage of 2/pi is adequate. -// -// Fetch P_1, P_2, P_3, P_4 beginning at bit position L thus: -// -// bit position L L-1 L-2 ... L-63 -// -// P_1 = b b b ... b -// -// each b can be 0 or 1. Also, let P_0 be the two bits corresponding to -// bit positions L+2 and L+1. So, when each of the P_j is interpreted -// with appropriate scaling, we have -// -// 2/pi = P_big + P_0 + (P_1 + P_2 + P_3 + P_4) + P_small -// -// Note that P_big and P_small can be ignored. The reasons are as follow. -// First, consider P_big. If P_big = 0, we can certainly ignore it. -// Otherwise, P_big >= 2^(L+3). Now, -// -// P_big * ulp(x) >= 2^(L+3) * 2^(m-63) -// >= 2^(65-m + m-63 ) -// >= 2^2 -// -// Thus, P_big * x is an integer of the form 4*K. So -// -// x = 4*K * (pi/2) + x*(P_0 + P_1 + P_2 + P_3 + P_4)*(pi/2) -// + x*P_small*(pi/2). -// -// Hence, P_big*x corresponds to information that can be ignored for -// trigonometic function evaluation. -// -// Next, we must estimate the effect of ignoring P_small. The absolute -// error made by ignoring P_small is bounded by -// -// |P_small * x| <= ulp(P_4) * x -// <= 2^(L-255) * 2^(m+1) -// <= 2^(62-m-255 + m + 1) -// <= 2^(-192) -// -// Since for double-extended precision, x * 2/pi = integer + f, -// 0.5 >= |f| >= 2^(-75), the relative error introduced by ignoring -// P_small is bounded by 2^(-192+75) <= 2^(-117), which is acceptable. -// -// Further note that if x is split into x_hi + x_lo where x_lo is the -// two bits corresponding to bit positions 2^(m-62) and 2^(m-63); then -// -// P_0 * x_hi -// -// is also an integer of the form 4*K; and thus can also be ignored. -// Let M := P_0 * x_lo which is a small integer. The main part of the -// calculation is really the multiplication of x with the four pieces -// P_1, P_2, P_3, and P_4. -// -// Unless the reduced argument is extremely small in magnitude, it -// suffices to carry out the multiplication of x with P_1, P_2, and -// P_3. x*P_4 will be carried out and added on as a correction only -// when it is found to be needed. Note also that x*P_4 need not be -// computed exactly. A straightforward multiplication suffices since -// the rounding error thus produced would be bounded by 2^(-3*64), -// that is 2^(-192) which is small enough as the reduced argument -// is bounded from below by 2^(-75). -// -// Now that we have four 64-bit data representing 2/pi and a -// 64-bit x. We first need to calculate a highly accurate product -// of x and P_1, P_2, P_3. This is best understood as integer -// multiplication. -// -// -// STEP 1. Multiplication -// ---------------------- -// -// -// --------- --------- --------- -// | P_1 | | P_2 | | P_3 | -// --------- --------- --------- -// -// --------- -// X | X | -// --------- -// ---------------------------------------------------- -// -// --------- --------- -// | A_hi | | A_lo | -// --------- --------- -// -// -// --------- --------- -// | B_hi | | B_lo | -// --------- --------- -// -// -// --------- --------- -// | C_hi | | C_lo | -// --------- --------- -// -// ==================================================== -// --------- --------- --------- --------- -// | S_0 | | S_1 | | S_2 | | S_3 | -// --------- --------- --------- --------- -// -// -// -// STEP 2. Get N and f -// ------------------- -// -// Conceptually, after the individual pieces S_0, S_1, ..., are obtained, -// we have to sum them and obtain an integer part, N, and a fraction, f. -// Here, |f| <= 1/2, and N is an integer. Note also that N need only to -// be known to module 2^k, k >= 2. In the case when |f| is small enough, -// we would need to add in the value x*P_4. -// -// -// STEP 3. Get reduced argument -// ---------------------------- -// -// The value f is not yet the reduced argument that we seek. The -// equation -// -// x * 2/pi = 4K + N + f -// -// says that -// -// x = 2*K*pi + N * pi/2 + f * (pi/2). -// -// Thus, the reduced argument is given by -// -// reduced argument = f * pi/2. -// -// This multiplication must be performed to extra precision. -// -// IV. Implementation -// ================== -// -// Step 0. Initialization -// ---------------------- -// -// Set sgn_x := sign(x); x := |x|; x_lo := 2 lsb of x. -// -// In memory, 2/pi is stored contiguously as -// -// 0x00000000 0x00000000 0xA2F.... -// ^ -// |__ implied binary bit -// -// Given x = 2^m * 1.xxxx...xxx; we calculate L := 62 - m. Thus -// -1 <= L <= -16321. We fetch from memory 5 integer pieces of data. -// -// P_0 is the two bits corresponding to bit positions L+2 and L+1 -// P_1 is the 64-bit starting at bit position L -// P_2 is the 64-bit starting at bit position L-64 -// P_3 is the 64-bit starting at bit position L-128 -// P_4 is the 64-bit starting at bit position L-192 -// -// For example, if m = 63, P_0 would be 0 and P_1 would look like -// 0xA2F... -// -// If m = 65, P_0 would be the two msb of 0xA, thus, P_0 is 10 in binary. -// P_1 in binary would be 1 0 0 0 1 0 1 1 1 1 .... -// -// Step 1. Multiplication -// ---------------------- -// -// At this point, P_1, P_2, P_3, P_4 are integers. They are -// supposed to be interpreted as -// -// 2^(L-63) * P_1; -// 2^(L-63-64) * P_2; -// 2^(L-63-128) * P_3; -// 2^(L-63-192) * P_4; -// -// Since each of them need to be multiplied to x, we would scale -// both x and the P_j's by some convenient factors: scale each -// of P_j's up by 2^(63-L), and scale x down by 2^(L-63). -// -// p_1 := fcvt.xf ( P_1 ) -// p_2 := fcvt.xf ( P_2 ) * 2^(-64) -// p_3 := fcvt.xf ( P_3 ) * 2^(-128) -// p_4 := fcvt.xf ( P_4 ) * 2^(-192) -// x := replace exponent of x by -1 -// because 2^m * 1.xxxx...xxx * 2^(L-63) -// is 2^(-1) * 1.xxxx...xxx -// -// We are now faced with the task of computing the following -// -// --------- --------- --------- -// | P_1 | | P_2 | | P_3 | -// --------- --------- --------- -// -// --------- -// X | X | -// --------- -// ---------------------------------------------------- -// -// --------- --------- -// | A_hi | | A_lo | -// --------- --------- -// -// --------- --------- -// | B_hi | | B_lo | -// --------- --------- -// -// --------- --------- -// | C_hi | | C_lo | -// --------- --------- -// -// ==================================================== -// ----------- --------- --------- --------- -// | S_0 | | S_1 | | S_2 | | S_3 | -// ----------- --------- --------- --------- -// ^ ^ -// | |___ binary point -// | -// |___ possibly one more bit -// -// Let FPSR3 be set to round towards zero with widest precision -// and exponent range. Unless an explicit FPSR is given, -// round-to-nearest with widest precision and exponent range is -// used. -// -// Define sigma_C := 2^63; sigma_B := 2^(-1); sigma_C := 2^(-65). -// -// Tmp_C := fmpy.fpsr3( x, p_1 ); -// If Tmp_C >= sigma_C then -// C_hi := Tmp_C; -// C_lo := x*p_1 - C_hi ...fma, exact -// Else -// C_hi := fadd.fpsr3(sigma_C, Tmp_C) - sigma_C -// ...subtraction is exact, regardless -// ...of rounding direction -// C_lo := x*p_1 - C_hi ...fma, exact -// End If -// -// Tmp_B := fmpy.fpsr3( x, p_2 ); -// If Tmp_B >= sigma_B then -// B_hi := Tmp_B; -// B_lo := x*p_2 - B_hi ...fma, exact -// Else -// B_hi := fadd.fpsr3(sigma_B, Tmp_B) - sigma_B -// ...subtraction is exact, regardless -// ...of rounding direction -// B_lo := x*p_2 - B_hi ...fma, exact -// End If -// -// Tmp_A := fmpy.fpsr3( x, p_3 ); -// If Tmp_A >= sigma_A then -// A_hi := Tmp_A; -// A_lo := x*p_3 - A_hi ...fma, exact -// Else -// A_hi := fadd.fpsr3(sigma_A, Tmp_A) - sigma_A -// ...subtraction is exact, regardless -// ...of rounding direction -// A_lo := x*p_3 - A_hi ...fma, exact -// End If -// -// ...Note that C_hi is of integer value. We need only the -// ...last few bits. Thus we can ensure C_hi is never a big -// ...integer, freeing us from overflow worry. -// -// Tmp_C := fadd.fpsr3( C_hi, 2^(70) ) - 2^(70); -// ...Tmp_C is the upper portion of C_hi -// C_hi := C_hi - Tmp_C -// ...0 <= C_hi < 2^7 -// -// Step 2. Get N and f -// ------------------- -// -// At this point, we have all the components to obtain -// S_0, S_1, S_2, S_3 and thus N and f. We start by adding -// C_lo and B_hi. This sum together with C_hi gives a good -// estimation of N and f. -// -// A := fadd.fpsr3( B_hi, C_lo ) -// B := max( B_hi, C_lo ) -// b := min( B_hi, C_lo ) -// -// a := (B - A) + b ...exact. Note that a is either 0 -// ...or 2^(-64). -// -// N := round_to_nearest_integer_value( A ); -// f := A - N; ...exact because lsb(A) >= 2^(-64) -// ...and |f| <= 1/2. -// -// f := f + a ...exact because a is 0 or 2^(-64); -// ...the msb of the sum is <= 1/2 -// ...lsb >= 2^(-64). -// -// N := convert to integer format( C_hi + N ); -// M := P_0 * x_lo; -// N := N + M; -// -// If sgn_x == 1 (that is original x was negative) -// N := 2^10 - N -// ...this maintains N to be non-negative, but still -// ...equivalent to the (negated N) mod 4. -// End If -// -// If |f| >= 2^(-33) -// -// ...Case 1 -// CASE := 1 -// g := A_hi + B_lo; -// s_hi := f + g; -// s_lo := (f - s_hi) + g; -// -// Else -// -// ...Case 2 -// CASE := 2 -// A := fadd.fpsr3( A_hi, B_lo ) -// B := max( A_hi, B_lo ) -// b := min( A_hi, B_lo ) -// -// a := (B - A) + b ...exact. Note that a is either 0 -// ...or 2^(-128). -// -// f_hi := A + f; -// f_lo := (f - f_hi) + A; -// ...this is exact. -// ...f-f_hi is exact because either |f| >= |A|, in which -// ...case f-f_hi is clearly exact; or otherwise, 0<|f|<|A| -// ...means msb(f) <= msb(A) = 2^(-64) => |f| = 2^(-64). -// ...If f = 2^(-64), f-f_hi involves cancellation and is -// ...exact. If f = -2^(-64), then A + f is exact. Hence -// ...f-f_hi is -A exactly, giving f_lo = 0. -// -// f_lo := f_lo + a; -// -// If |f| >= 2^(-50) then -// s_hi := f_hi; -// s_lo := f_lo; -// Else -// f_lo := (f_lo + A_lo) + x*p_4 -// s_hi := f_hi + f_lo -// s_lo := (f_hi - s_hi) + f_lo -// End If -// -// End If -// -// Step 3. Get reduced argument -// ---------------------------- -// -// If sgn_x == 0 (that is original x is positive) -// -// D_hi := Pi_by_2_hi -// D_lo := Pi_by_2_lo -// ...load from table -// -// Else -// -// D_hi := neg_Pi_by_2_hi -// D_lo := neg_Pi_by_2_lo -// ...load from table -// End If -// -// r_hi := s_hi*D_hi -// r_lo := s_hi*D_hi - r_hi ...fma -// r_lo := (s_hi*D_lo + r_lo) + s_lo*D_hi -// -// Return N, r_hi, r_lo -// -FR_input_X = f8 -FR_r_hi = f8 -FR_r_lo = f9 - -FR_X = f32 -FR_N = f33 -FR_p_1 = f34 -FR_TWOM33 = f35 -FR_TWOM50 = f36 -FR_g = f37 -FR_p_2 = f38 -FR_f = f39 -FR_s_lo = f40 -FR_p_3 = f41 -FR_f_abs = f42 -FR_D_lo = f43 -FR_p_4 = f44 -FR_D_hi = f45 -FR_Tmp2_C = f46 -FR_s_hi = f47 -FR_sigma_A = f48 -FR_A = f49 -FR_sigma_B = f50 -FR_B = f51 -FR_sigma_C = f52 -FR_b = f53 -FR_ScaleP2 = f54 -FR_ScaleP3 = f55 -FR_ScaleP4 = f56 -FR_Tmp_A = f57 -FR_Tmp_B = f58 -FR_Tmp_C = f59 -FR_A_hi = f60 -FR_f_hi = f61 -FR_RSHF = f62 -FR_A_lo = f63 -FR_B_hi = f64 -FR_a = f65 -FR_B_lo = f66 -FR_f_lo = f67 -FR_N_fix = f68 -FR_C_hi = f69 -FR_C_lo = f70 - -GR_N = r8 -GR_Exp_x = r36 -GR_Temp = r37 -GR_BIASL63 = r38 -GR_CASE = r39 -GR_x_lo = r40 -GR_sgn_x = r41 -GR_M = r42 -GR_BASE = r43 -GR_LENGTH1 = r44 -GR_LENGTH2 = r45 -GR_ASUB = r46 -GR_P_0 = r47 -GR_P_1 = r48 -GR_P_2 = r49 -GR_P_3 = r50 -GR_P_4 = r51 -GR_START = r52 -GR_SEGMENT = r53 -GR_A = r54 -GR_B = r55 -GR_C = r56 -GR_D = r57 -GR_E = r58 -GR_TEMP1 = r59 -GR_TEMP2 = r60 -GR_TEMP3 = r61 -GR_TEMP4 = r62 -GR_TEMP5 = r63 -GR_TEMP6 = r64 -GR_rshf = r64 - -RODATA -.align 64 - -LOCAL_OBJECT_START(Constants_Bits_of_2_by_pi) -data8 0x0000000000000000,0xA2F9836E4E441529 -data8 0xFC2757D1F534DDC0,0xDB6295993C439041 -data8 0xFE5163ABDEBBC561,0xB7246E3A424DD2E0 -data8 0x06492EEA09D1921C,0xFE1DEB1CB129A73E -data8 0xE88235F52EBB4484,0xE99C7026B45F7E41 -data8 0x3991D639835339F4,0x9C845F8BBDF9283B -data8 0x1FF897FFDE05980F,0xEF2F118B5A0A6D1F -data8 0x6D367ECF27CB09B7,0x4F463F669E5FEA2D -data8 0x7527BAC7EBE5F17B,0x3D0739F78A5292EA 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0x921110D8E80FAF80,0x6C4BFFDB0F903876 -data8 0x185915A562BBCB61,0xB989C7BD401004F2 -data8 0xD2277549F6B6EBBB,0x22DBAA140A2F2689 -data8 0x768364333B091A94,0x0EAA3A51C2A31DAE -data8 0xEDAF12265C4DC26D,0x9C7A2D9756C0833F -data8 0x03F6F0098C402B99,0x316D07B43915200C -data8 0x5BC3D8C492F54BAD,0xC6A5CA4ECD37A736 -data8 0xA9E69492AB6842DD,0xDE6319EF8C76528B -data8 0x6837DBFCABA1AE31,0x15DFA1AE00DAFB0C -data8 0x664D64B705ED3065,0x29BF56573AFF47B9 -data8 0xF96AF3BE75DF9328,0x3080ABF68C6615CB -data8 0x040622FA1DE4D9A4,0xB33D8F1B5709CD36 -data8 0xE9424EA4BE13B523,0x331AAAF0A8654FA5 -data8 0xC1D20F3F0BCD785B,0x76F923048B7B7217 -data8 0x8953A6C6E26E6F00,0xEBEF584A9BB7DAC4 -data8 0xBA66AACFCF761D02,0xD12DF1B1C1998C77 -data8 0xADC3DA4886A05DF7,0xF480C62FF0AC9AEC -data8 0xDDBC5C3F6DDED01F,0xC790B6DB2A3A25A3 -data8 0x9AAF009353AD0457,0xB6B42D297E804BA7 -data8 0x07DA0EAA76A1597B,0x2A12162DB7DCFDE5 -data8 0xFAFEDB89FDBE896C,0x76E4FCA90670803E -data8 0x156E85FF87FD073E,0x2833676186182AEA -data8 0xBD4DAFE7B36E6D8F,0x3967955BBF3148D7 -data8 0x8416DF30432DC735,0x6125CE70C9B8CB30 -data8 0xFD6CBFA200A4E46C,0x05A0DD5A476F21D2 -data8 0x1262845CB9496170,0xE0566B0152993755 -data8 0x50B7D51EC4F1335F,0x6E13E4305DA92E85 -data8 0xC3B21D3632A1A4B7,0x08D4B1EA21F716E4 -data8 0x698F77FF2780030C,0x2D408DA0CD4F99A5 -data8 0x20D3A2B30A5D2F42,0xF9B4CBDA11D0BE7D -data8 0xC1DB9BBD17AB81A2,0xCA5C6A0817552E55 -data8 0x0027F0147F8607E1,0x640B148D4196DEBE -data8 0x872AFDDAB6256B34,0x897BFEF3059EBFB9 -data8 0x4F6A68A82A4A5AC4,0x4FBCF82D985AD795 -data8 0xC7F48D4D0DA63A20,0x5F57A4B13F149538 -data8 0x800120CC86DD71B6,0xDEC9F560BF11654D -data8 0x6B0701ACB08CD0C0,0xB24855510EFB1EC3 -data8 0x72953B06A33540C0,0x7BDC06CC45E0FA29 -data8 0x4EC8CAD641F3E8DE,0x647CD8649B31BED9 -data8 0xC397A4D45877C5E3,0x6913DAF03C3ABA46 -data8 0x18465F7555F5BDD2,0xC6926E5D2EACED44 -data8 0x0E423E1C87C461E9,0xFD29F3D6E7CA7C22 -data8 0x35916FC5E0088DD7,0xFFE26A6EC6FDB0C1 -data8 0x0893745D7CB2AD6B,0x9D6ECD7B723E6A11 -data8 0xC6A9CFF7DF7329BA,0xC9B55100B70DB2E2 -data8 0x24BA74607DE58AD8,0x742C150D0C188194 -data8 0x667E162901767A9F,0xBEFDFDEF4556367E -data8 0xD913D9ECB9BA8BFC,0x97C427A831C36EF1 -data8 0x36C59456A8D8B5A8,0xB40ECCCF2D891234 -data8 0x576F89562CE3CE99,0xB920D6AA5E6B9C2A -data8 0x3ECC5F114A0BFDFB,0xF4E16D3B8E2C86E2 -data8 0x84D4E9A9B4FCD1EE,0xEFC9352E61392F44 -data8 0x2138C8D91B0AFC81,0x6A4AFBD81C2F84B4 -data8 0x538C994ECC2254DC,0x552AD6C6C096190B -data8 0xB8701A649569605A,0x26EE523F0F117F11 -data8 0xB5F4F5CBFC2DBC34,0xEEBC34CC5DE8605E -data8 0xDD9B8E67EF3392B8,0x17C99B5861BC57E1 -data8 0xC68351103ED84871,0xDDDD1C2DA118AF46 -data8 0x2C21D7F359987AD9,0xC0549EFA864FFC06 -data8 0x56AE79E536228922,0xAD38DC9367AAE855 -data8 0x3826829BE7CAA40D,0x51B133990ED7A948 -data8 0x0569F0B265A7887F,0x974C8836D1F9B392 -data8 0x214A827B21CF98DC,0x9F405547DC3A74E1 -data8 0x42EB67DF9DFE5FD4,0x5EA4677B7AACBAA2 -data8 0xF65523882B55BA41,0x086E59862A218347 -data8 0x39E6E389D49EE540,0xFB49E956FFCA0F1C -data8 0x8A59C52BFA94C5C1,0xD3CFC50FAE5ADB86 -data8 0xC5476243853B8621,0x94792C8761107B4C -data8 0x2A1A2C8012BF4390,0x2688893C78E4C4A8 -data8 0x7BDBE5C23AC4EAF4,0x268A67F7BF920D2B -data8 0xA365B1933D0B7CBD,0xDC51A463DD27DDE1 -data8 0x6919949A9529A828,0xCE68B4ED09209F44 -data8 0xCA984E638270237C,0x7E32B90F8EF5A7E7 -data8 0x561408F1212A9DB5,0x4D7E6F5119A5ABF9 -data8 0xB5D6DF8261DD9602,0x36169F3AC4A1A283 -data8 0x6DED727A8D39A9B8,0x825C326B5B2746ED -data8 0x34007700D255F4FC,0x4D59018071E0E13F -data8 0x89B295F364A8F1AE,0xA74B38FC4CEAB2BB -LOCAL_OBJECT_END(Constants_Bits_of_2_by_pi) - -LOCAL_OBJECT_START(Constants_Bits_of_pi_by_2) -data8 0xC90FDAA22168C234,0x00003FFF -data8 0xC4C6628B80DC1CD1,0x00003FBF -LOCAL_OBJECT_END(Constants_Bits_of_pi_by_2) - -.section .text -.global __libm_pi_by_2_reduce# -.proc __libm_pi_by_2_reduce# -.align 32 - -__libm_pi_by_2_reduce: - -// X is in f8 -// Place the two-piece result r (r_hi) in f8 and c (r_lo) in f9 -// N is returned in r8 - -{ .mfi - alloc r34 = ar.pfs,2,34,0,0 - fsetc.s3 0x00,0x7F // Set sf3 to round to zero, 82-bit prec, td, ftz - nop.i 999 -} -{ .mfi - addl GR_BASE = @ltoff(Constants_Bits_of_2_by_pi#), gp - nop.f 999 - mov GR_BIASL63 = 0x1003E -} -;; - - -// L -1-2-3-4 -// 0 0 0 0 0. 1 0 1 0 -// M 0 1 2 .... 63, 64 65 ... 127, 128 -// --------------------------------------------- -// Segment 0. 1 , 2 , 3 -// START = M - 63 M = 128 becomes 65 -// LENGTH1 = START & 0x3F 65 become position 1 -// SEGMENT = shr(START,6) + 1 0 maps to 1, 64 maps to 2, -// LENGTH2 = 64 - LENGTH1 -// Address_BASE = shladd(SEGMENT,3) + BASE - - -{ .mmi - getf.exp GR_Exp_x = FR_input_X - ld8 GR_BASE = [GR_BASE] - mov GR_TEMP5 = 0x0FFFE -} -;; - -// Define sigma_C := 2^63; sigma_B := 2^(-1); sigma_A := 2^(-65). -{ .mmi - getf.sig GR_x_lo = FR_input_X - mov GR_TEMP6 = 0x0FFBE - nop.i 999 -} -;; - -// Special Code for testing DE arguments -// movl GR_BIASL63 = 0x0000000000013FFE -// movl GR_x_lo = 0xFFFFFFFFFFFFFFFF -// setf.exp FR_X = GR_BIASL63 -// setf.sig FR_ScaleP3 = GR_x_lo -// fmerge.se FR_X = FR_X,FR_ScaleP3 -// Set sgn_x := sign(x); x := |x|; x_lo := 2 lsb of x. -// 2/pi is stored contiguously as -// 0x00000000 0x00000000.0xA2F.... -// M = EXP - BIAS ( M >= 63) -// Given x = 2^m * 1.xxxx...xxx; we calculate L := 62 - m. -// Thus -1 <= L <= -16321. -{ .mmi - setf.exp FR_sigma_B = GR_TEMP5 - setf.exp FR_sigma_A = GR_TEMP6 - extr.u GR_M = GR_Exp_x,0,17 -} -;; - -{ .mii - and GR_x_lo = 0x03,GR_x_lo - sub GR_START = GR_M,GR_BIASL63 - add GR_BASE = 8,GR_BASE // To effectively add 1 to SEGMENT -} -;; - -{ .mii - and GR_LENGTH1 = 0x3F,GR_START - shr.u GR_SEGMENT = GR_START,6 - nop.i 999 -} -;; - -{ .mmi - shladd GR_BASE = GR_SEGMENT,3,GR_BASE - sub GR_LENGTH2 = 0x40,GR_LENGTH1 - cmp.le p6,p7 = 0x2,GR_LENGTH1 -} -;; - -// P_0 is the two bits corresponding to bit positions L+2 and L+1 -// P_1 is the 64-bit starting at bit position L -// P_2 is the 64-bit starting at bit position L-64 -// P_3 is the 64-bit starting at bit position L-128 -// P_4 is the 64-bit starting at bit position L-192 -// P_1 is made up of Alo and Bhi -// P_1 = deposit Alo, position 0, length2 into P_1,position length1 -// deposit Bhi, position length2, length1 into P_1, position 0 -// P_2 is made up of Blo and Chi -// P_2 = deposit Blo, position 0, length2 into P_2, position length1 -// deposit Chi, position length2, length1 into P_2, position 0 -// P_3 is made up of Clo and Dhi -// P_3 = deposit Clo, position 0, length2 into P_3, position length1 -// deposit Dhi, position length2, length1 into P_3, position 0 -// P_4 is made up of Clo and Dhi -// P_4 = deposit Dlo, position 0, length2 into P_4, position length1 -// deposit Ehi, position length2, length1 into P_4, position 0 -{ .mfi - ld8 GR_A = [GR_BASE],8 - fabs FR_X = FR_input_X -(p7) cmp.eq.unc p8,p9 = 0x1,GR_LENGTH1 -} -;; - -// ld_64 A at Base and increment Base by 8 -// ld_64 B at Base and increment Base by 8 -// ld_64 C at Base and increment Base by 8 -// ld_64 D at Base and increment Base by 8 -// ld_64 E at Base and increment Base by 8 -// A/B/C/D -// --------------------- -// A, B, C, D, and E look like | length1 | length2 | -// --------------------- -// hi lo -{ .mlx - ld8 GR_B = [GR_BASE],8 - movl GR_rshf = 0x43e8000000000000 // 1.10000 2^63 for right shift N_fix -} -;; - -{ .mmi - ld8 GR_C = [GR_BASE],8 - nop.m 999 -(p8) extr.u GR_Temp = GR_A,63,1 -} -;; - -// If length1 >= 2, -// P_0 = deposit Ahi, position length2, 2 bit into P_0 at position 0. -{ .mii - ld8 GR_D = [GR_BASE],8 - shl GR_TEMP1 = GR_A,GR_LENGTH1 // MM instruction -(p6) shr.u GR_P_0 = GR_A,GR_LENGTH2 // MM instruction -} -;; - -{ .mii - ld8 GR_E = [GR_BASE],-40 - shl GR_TEMP2 = GR_B,GR_LENGTH1 // MM instruction - shr.u GR_P_1 = GR_B,GR_LENGTH2 // MM instruction -} -;; - -// Else -// Load 16 bit of ASUB from (Base_Address_of_A - 2) -// P_0 = ASUB & 0x3 -// If length1 == 0, -// P_0 complete -// Else -// Deposit element 63 from Ahi and place in element 0 of P_0. -// Endif -// Endif - -{ .mii -(p7) ld2 GR_ASUB = [GR_BASE],8 - shl GR_TEMP3 = GR_C,GR_LENGTH1 // MM instruction - shr.u GR_P_2 = GR_C,GR_LENGTH2 // MM instruction -} -;; - -{ .mii - setf.d FR_RSHF = GR_rshf // Form right shift const 1.100 * 2^63 - shl GR_TEMP4 = GR_D,GR_LENGTH1 // MM instruction - shr.u GR_P_3 = GR_D,GR_LENGTH2 // MM instruction -} -;; - -{ .mmi -(p7) and GR_P_0 = 0x03,GR_ASUB -(p6) and GR_P_0 = 0x03,GR_P_0 - shr.u GR_P_4 = GR_E,GR_LENGTH2 // MM instruction -} -;; - -{ .mmi - nop.m 999 - or GR_P_1 = GR_P_1,GR_TEMP1 -(p8) and GR_P_0 = 0x1,GR_P_0 -} -;; - -{ .mmi - setf.sig FR_p_1 = GR_P_1 - or GR_P_2 = GR_P_2,GR_TEMP2 -(p8) shladd GR_P_0 = GR_P_0,1,GR_Temp -} -;; - -{ .mmf - setf.sig FR_p_2 = GR_P_2 - or GR_P_3 = GR_P_3,GR_TEMP3 - fmerge.se FR_X = FR_sigma_B,FR_X -} -;; - -{ .mmi - setf.sig FR_p_3 = GR_P_3 - or GR_P_4 = GR_P_4,GR_TEMP4 - pmpy2.r GR_M = GR_P_0,GR_x_lo -} -;; - -// P_1, P_2, P_3, P_4 are integers. They should be -// 2^(L-63) * P_1; -// 2^(L-63-64) * P_2; -// 2^(L-63-128) * P_3; -// 2^(L-63-192) * P_4; -// Since each of them need to be multiplied to x, we would scale -// both x and the P_j's by some convenient factors: scale each -// of P_j's up by 2^(63-L), and scale x down by 2^(L-63). -// p_1 := fcvt.xf ( P_1 ) -// p_2 := fcvt.xf ( P_2 ) * 2^(-64) -// p_3 := fcvt.xf ( P_3 ) * 2^(-128) -// p_4 := fcvt.xf ( P_4 ) * 2^(-192) -// x= Set x's exp to -1 because 2^m*1.x...x *2^(L-63)=2^(-1)*1.x...xxx -// --------- --------- --------- -// | P_1 | | P_2 | | P_3 | -// --------- --------- --------- -// --------- -// X | X | -// --------- -// ---------------------------------------------------- -// --------- --------- -// | A_hi | | A_lo | -// --------- --------- -// --------- --------- -// | B_hi | | B_lo | -// --------- --------- -// --------- --------- -// | C_hi | | C_lo | -// --------- --------- -// ==================================================== -// ----------- --------- --------- --------- -// | S_0 | | S_1 | | S_2 | | S_3 | -// ----------- --------- --------- --------- -// | |___ binary point -// |___ possibly one more bit -// -// Let FPSR3 be set to round towards zero with widest precision -// and exponent range. Unless an explicit FPSR is given, -// round-to-nearest with widest precision and exponent range is -// used. -{ .mmi - setf.sig FR_p_4 = GR_P_4 - mov GR_TEMP1 = 0x0FFBF - nop.i 999 -} -;; - -{ .mmi - setf.exp FR_ScaleP2 = GR_TEMP1 - mov GR_TEMP2 = 0x0FF7F - nop.i 999 -} -;; - -{ .mmi - setf.exp FR_ScaleP3 = GR_TEMP2 - mov GR_TEMP4 = 0x1003E - nop.i 999 -} -;; - -{ .mmf - setf.exp FR_sigma_C = GR_TEMP4 - mov GR_Temp = 0x0FFDE - fcvt.xuf.s1 FR_p_1 = FR_p_1 -} -;; - -{ .mfi - setf.exp FR_TWOM33 = GR_Temp - fcvt.xuf.s1 FR_p_2 = FR_p_2 - nop.i 999 -} -;; - -{ .mfi - nop.m 999 - fcvt.xuf.s1 FR_p_3 = FR_p_3 - nop.i 999 -} -;; - -{ .mfi - nop.m 999 - fcvt.xuf.s1 FR_p_4 = FR_p_4 - nop.i 999 -} -;; - -// Tmp_C := fmpy.fpsr3( x, p_1 ); -// Tmp_B := fmpy.fpsr3( x, p_2 ); -// Tmp_A := fmpy.fpsr3( x, p_3 ); -// If Tmp_C >= sigma_C then -// C_hi := Tmp_C; -// C_lo := x*p_1 - C_hi ...fma, exact -// Else -// C_hi := fadd.fpsr3(sigma_C, Tmp_C) - sigma_C -// C_lo := x*p_1 - C_hi ...fma, exact -// End If -// If Tmp_B >= sigma_B then -// B_hi := Tmp_B; -// B_lo := x*p_2 - B_hi ...fma, exact -// Else -// B_hi := fadd.fpsr3(sigma_B, Tmp_B) - sigma_B -// B_lo := x*p_2 - B_hi ...fma, exact -// End If -// If Tmp_A >= sigma_A then -// A_hi := Tmp_A; -// A_lo := x*p_3 - A_hi ...fma, exact -// Else -// A_hi := fadd.fpsr3(sigma_A, Tmp_A) - sigma_A -// Exact, regardless ...of rounding direction -// A_lo := x*p_3 - A_hi ...fma, exact -// Endif -{ .mfi - nop.m 999 - fmpy.s3 FR_Tmp_C = FR_X,FR_p_1 - nop.i 999 -} -;; - -{ .mfi - mov GR_TEMP3 = 0x0FF3F - fmpy.s1 FR_p_2 = FR_p_2,FR_ScaleP2 - nop.i 999 -} -;; - -{ .mmf - setf.exp FR_ScaleP4 = GR_TEMP3 - mov GR_TEMP4 = 0x10045 - fmpy.s1 FR_p_3 = FR_p_3,FR_ScaleP3 -} -;; - -{ .mfi - nop.m 999 - fadd.s3 FR_C_hi = FR_sigma_C,FR_Tmp_C // For Tmp_C < sigma_C case - nop.i 999 -} -;; - -{ .mmf - setf.exp FR_Tmp2_C = GR_TEMP4 - nop.m 999 - fmpy.s3 FR_Tmp_B = FR_X,FR_p_2 -} -;; - -{ .mfi - addl GR_BASE = @ltoff(Constants_Bits_of_pi_by_2#), gp - fcmp.ge.s1 p12, p9 = FR_Tmp_C,FR_sigma_C - nop.i 999 -} -{ .mfi - nop.m 999 - fmpy.s3 FR_Tmp_A = FR_X,FR_p_3 - nop.i 99 -} -;; - -{ .mfi - ld8 GR_BASE = [GR_BASE] -(p12) mov FR_C_hi = FR_Tmp_C - nop.i 999 -} -{ .mfi - nop.m 999 -(p9) fsub.s1 FR_C_hi = FR_C_hi,FR_sigma_C - nop.i 999 -} -;; - - - -// End If -// Step 3. Get reduced argument -// If sgn_x == 0 (that is original x is positive) -// D_hi := Pi_by_2_hi -// D_lo := Pi_by_2_lo -// Load from table -// Else -// D_hi := neg_Pi_by_2_hi -// D_lo := neg_Pi_by_2_lo -// Load from table -// End If - -{ .mfi - nop.m 999 - fmpy.s1 FR_p_4 = FR_p_4,FR_ScaleP4 - nop.i 999 -} -{ .mfi - nop.m 999 - fadd.s3 FR_B_hi = FR_sigma_B,FR_Tmp_B // For Tmp_B < sigma_B case - nop.i 999 -} -;; - -{ .mfi - nop.m 999 - fadd.s3 FR_A_hi = FR_sigma_A,FR_Tmp_A // For Tmp_A < sigma_A case - nop.i 999 -} -;; - -{ .mfi - nop.m 999 - fcmp.ge.s1 p13, p10 = FR_Tmp_B,FR_sigma_B - nop.i 999 -} -{ .mfi - nop.m 999 - fms.s1 FR_C_lo = FR_X,FR_p_1,FR_C_hi - nop.i 999 -} -;; - -{ .mfi - ldfe FR_D_hi = [GR_BASE],16 - fcmp.ge.s1 p14, p11 = FR_Tmp_A,FR_sigma_A - nop.i 999 -} -;; - -{ .mfi - ldfe FR_D_lo = [GR_BASE] -(p13) mov FR_B_hi = FR_Tmp_B - nop.i 999 -} -{ .mfi - nop.m 999 -(p10) fsub.s1 FR_B_hi = FR_B_hi,FR_sigma_B - nop.i 999 -} -;; - -{ .mfi - nop.m 999 -(p14) mov FR_A_hi = FR_Tmp_A - nop.i 999 -} -{ .mfi - nop.m 999 -(p11) fsub.s1 FR_A_hi = FR_A_hi,FR_sigma_A - nop.i 999 -} -;; - -// Note that C_hi is of integer value. We need only the -// last few bits. Thus we can ensure C_hi is never a big -// integer, freeing us from overflow worry. -// Tmp_C := fadd.fpsr3( C_hi, 2^(70) ) - 2^(70); -// Tmp_C is the upper portion of C_hi -{ .mfi - nop.m 999 - fadd.s3 FR_Tmp_C = FR_C_hi,FR_Tmp2_C - tbit.z p12,p9 = GR_Exp_x, 17 -} -;; - -{ .mfi - nop.m 999 - fms.s1 FR_B_lo = FR_X,FR_p_2,FR_B_hi - nop.i 999 -} -{ .mfi - nop.m 999 - fadd.s3 FR_A = FR_B_hi,FR_C_lo - nop.i 999 -} -;; - -{ .mfi - nop.m 999 - fms.s1 FR_A_lo = FR_X,FR_p_3,FR_A_hi - nop.i 999 -} -;; - -{ .mfi - nop.m 999 - fsub.s1 FR_Tmp_C = FR_Tmp_C,FR_Tmp2_C - nop.i 999 -} -;; - -// ******************* -// Step 2. Get N and f -// ******************* -// We have all the components to obtain -// S_0, S_1, S_2, S_3 and thus N and f. We start by adding -// C_lo and B_hi. This sum together with C_hi estimates -// N and f well. -// A := fadd.fpsr3( B_hi, C_lo ) -// B := max( B_hi, C_lo ) -// b := min( B_hi, C_lo ) -{ .mfi - nop.m 999 - fmax.s1 FR_B = FR_B_hi,FR_C_lo - nop.i 999 -} -;; - -// We use a right-shift trick to get the integer part of A into the rightmost -// bits of the significand by adding 1.1000..00 * 2^63. This operation is good -// if |A| < 2^61, which it is in this case. We are doing this to save a few -// cycles over using fcvt.fx followed by fnorm. The second step of the trick -// is to subtract the same constant to float the rounded integer into a fp reg. - -{ .mfi - nop.m 999 -// N := round_to_nearest_integer_value( A ); - fma.s1 FR_N_fix = FR_A, f1, FR_RSHF - nop.i 999 -} -;; - -{ .mfi - nop.m 999 - fmin.s1 FR_b = FR_B_hi,FR_C_lo - nop.i 999 -} -{ .mfi - nop.m 999 -// C_hi := C_hi - Tmp_C ...0 <= C_hi < 2^7 - fsub.s1 FR_C_hi = FR_C_hi,FR_Tmp_C - nop.i 999 -} -;; - -{ .mfi - nop.m 999 -// a := (B - A) + b: Exact - note that a is either 0 or 2^(-64). - fsub.s1 FR_a = FR_B,FR_A - nop.i 999 -} -;; - -{ .mfi - nop.m 999 - fms.s1 FR_N = FR_N_fix, f1, FR_RSHF - nop.i 999 -} -;; - -{ .mfi - nop.m 999 - fadd.s1 FR_a = FR_a,FR_b - nop.i 999 -} -;; - -// f := A - N; Exact because lsb(A) >= 2^(-64) and |f| <= 1/2. -// N := convert to integer format( C_hi + N ); -// M := P_0 * x_lo; -// N := N + M; -{ .mfi - nop.m 999 - fsub.s1 FR_f = FR_A,FR_N - nop.i 999 -} -{ .mfi - nop.m 999 - fadd.s1 FR_N = FR_N,FR_C_hi - nop.i 999 -} -;; - -{ .mfi - nop.m 999 -(p9) fsub.s1 FR_D_hi = f0, FR_D_hi - nop.i 999 -} -{ .mfi - nop.m 999 -(p9) fsub.s1 FR_D_lo = f0, FR_D_lo - nop.i 999 -} -;; - -{ .mfi - nop.m 999 - fadd.s1 FR_g = FR_A_hi,FR_B_lo // For Case 1, g=A_hi+B_lo - nop.i 999 -} -{ .mfi - nop.m 999 - fadd.s3 FR_A = FR_A_hi,FR_B_lo // For Case 2, A=A_hi+B_lo w/ sf3 - nop.i 999 -} -;; - -{ .mfi - mov GR_Temp = 0x0FFCD // For Case 2, exponent of 2^-50 - fmax.s1 FR_B = FR_A_hi,FR_B_lo // For Case 2, B=max(A_hi,B_lo) - nop.i 999 -} -;; - -// f = f + a Exact because a is 0 or 2^(-64); -// the msb of the sum is <= 1/2 and lsb >= 2^(-64). -{ .mfi - setf.exp FR_TWOM50 = GR_Temp // For Case 2, form 2^-50 - fcvt.fx.s1 FR_N = FR_N - nop.i 999 -} -{ .mfi - nop.m 999 - fadd.s1 FR_f = FR_f,FR_a - nop.i 999 -} -;; - -{ .mfi - nop.m 999 - fmin.s1 FR_b = FR_A_hi,FR_B_lo // For Case 2, b=min(A_hi,B_lo) - nop.i 999 -} -;; - -{ .mfi - nop.m 999 - fsub.s1 FR_a = FR_B,FR_A // For Case 2, a=B-A - nop.i 999 -} -;; - -{ .mfi - nop.m 999 - fadd.s1 FR_s_hi = FR_f,FR_g // For Case 1, s_hi=f+g - nop.i 999 -} -{ .mfi - nop.m 999 - fadd.s1 FR_f_hi = FR_A,FR_f // For Case 2, f_hi=A+f - nop.i 999 -} -;; - -{ .mfi - nop.m 999 - fabs FR_f_abs = FR_f - nop.i 999 -} -;; - -{ .mfi - getf.sig GR_N = FR_N - fsetc.s3 0x7F,0x40 // Reset sf3 to user settings + td - nop.i 999 -} -;; - -{ .mfi - nop.m 999 - fsub.s1 FR_s_lo = FR_f,FR_s_hi // For Case 1, s_lo=f-s_hi - nop.i 999 -} -{ .mfi - nop.m 999 - fsub.s1 FR_f_lo = FR_f,FR_f_hi // For Case 2, f_lo=f-f_hi - nop.i 999 -} -;; - -{ .mfi - nop.m 999 - fmpy.s1 FR_r_hi = FR_s_hi,FR_D_hi // For Case 1, r_hi=s_hi*D_hi - nop.i 999 -} -{ .mfi - nop.m 999 - fadd.s1 FR_a = FR_a,FR_b // For Case 2, a=a+b - nop.i 999 -} -;; - - -// If sgn_x == 1 (that is original x was negative) -// N := 2^10 - N -// this maintains N to be non-negative, but still -// equivalent to the (negated N) mod 4. -// End If -{ .mfi - add GR_N = GR_N,GR_M - fcmp.ge.s1 p13, p10 = FR_f_abs,FR_TWOM33 - mov GR_Temp = 0x00400 -} -;; - -{ .mfi -(p9) sub GR_N = GR_Temp,GR_N - fadd.s1 FR_s_lo = FR_s_lo,FR_g // For Case 1, s_lo=s_lo+g - nop.i 999 -} -{ .mfi - nop.m 999 - fadd.s1 FR_f_lo = FR_f_lo,FR_A // For Case 2, f_lo=f_lo+A - nop.i 999 -} -;; - -// a := (B - A) + b Exact. -// Note that a is either 0 or 2^(-128). -// f_hi := A + f; -// f_lo := (f - f_hi) + A -// f_lo=f-f_hi is exact because either |f| >= |A|, in which -// case f-f_hi is clearly exact; or otherwise, 0<|f|<|A| -// means msb(f) <= msb(A) = 2^(-64) => |f| = 2^(-64). -// If f = 2^(-64), f-f_hi involves cancellation and is -// exact. If f = -2^(-64), then A + f is exact. Hence -// f-f_hi is -A exactly, giving f_lo = 0. -// f_lo := f_lo + a; - -// If |f| >= 2^(-33) -// Case 1 -// CASE := 1 -// g := A_hi + B_lo; -// s_hi := f + g; -// s_lo := (f - s_hi) + g; -// Else -// Case 2 -// CASE := 2 -// A := fadd.fpsr3( A_hi, B_lo ) -// B := max( A_hi, B_lo ) -// b := min( A_hi, B_lo ) - -{ .mfi - nop.m 999 -(p10) fcmp.ge.unc.s1 p14, p11 = FR_f_abs,FR_TWOM50 - nop.i 999 -} -{ .mfi - nop.m 999 -(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 - nop.i 999 -} -;; - -// If |f| >= 2^(-50) then -// s_hi := f_hi; -// s_lo := f_lo; -// Else -// f_lo := (f_lo + A_lo) + x*p_4 -// s_hi := f_hi + f_lo -// s_lo := (f_hi - s_hi) + f_lo -// End If -{ .mfi - nop.m 999 -(p14) mov FR_s_hi = FR_f_hi - nop.i 999 -} -{ .mfi - nop.m 999 -(p10) fadd.s1 FR_f_lo = FR_f_lo,FR_a - nop.i 999 -} -;; - -{ .mfi - nop.m 999 -(p14) mov FR_s_lo = FR_f_lo - nop.i 999 -} -{ .mfi - nop.m 999 -(p11) fadd.s1 FR_f_lo = FR_f_lo,FR_A_lo - nop.i 999 -} -;; - -{ .mfi - nop.m 999 -(p11) fma.s1 FR_f_lo = FR_X,FR_p_4,FR_f_lo - nop.i 999 -} -;; - -{ .mfi - nop.m 999 -(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 - nop.i 999 -} -{ .mfi - nop.m 999 -(p11) fadd.s1 FR_s_hi = FR_f_hi,FR_f_lo - nop.i 999 -} -;; - -// r_hi := s_hi*D_hi -// r_lo := s_hi*D_hi - r_hi with fma -// r_lo := (s_hi*D_lo + r_lo) + s_lo*D_hi -{ .mfi - nop.m 999 -(p10) fmpy.s1 FR_r_hi = FR_s_hi,FR_D_hi - nop.i 999 -} -{ .mfi - nop.m 999 -(p11) fsub.s1 FR_s_lo = FR_f_hi,FR_s_hi - nop.i 999 -} -;; - -{ .mfi - nop.m 999 -(p10) fms.s1 FR_r_lo = FR_s_hi,FR_D_hi,FR_r_hi - nop.i 999 -} -{ .mfi - nop.m 999 -(p11) fadd.s1 FR_s_lo = FR_s_lo,FR_f_lo - nop.i 999 -} -;; - -{ .mfi - nop.m 999 -(p10) fma.s1 FR_r_lo = FR_s_hi,FR_D_lo,FR_r_lo - nop.i 999 -} -;; - -// Return N, r_hi, r_lo -// We do not return CASE -{ .mfb - nop.m 999 - fma.s1 FR_r_lo = FR_s_lo,FR_D_hi,FR_r_lo - br.ret.sptk b0 -} -;; - -.endp __libm_pi_by_2_reduce# |