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Diffstat (limited to 'sysdeps/ia64/fpu/libm_reduce.S')
| -rw-r--r-- | sysdeps/ia64/fpu/libm_reduce.S | 1578 |
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diff --git a/sysdeps/ia64/fpu/libm_reduce.S b/sysdeps/ia64/fpu/libm_reduce.S new file mode 100644 index 0000000..8b13249 --- /dev/null +++ b/sysdeps/ia64/fpu/libm_reduce.S @@ -0,0 +1,1578 @@ +.file "libm_reduce.s" + + +// Copyright (c) 2000 - 2003, Intel Corporation +// All rights reserved. +// +// Contributed 2000 by the Intel Numerics Group, Intel Corporation +// +// 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 interger 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 correspoding 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 +data8 0x6BFB5FB11F8D5D08,0x56033046FC7B6BAB +data8 0xF0CFBC209AF4361D,0xA9E391615EE61B08 +data8 0x6599855F14A06840,0x8DFFD8804D732731 +data8 0x06061556CA73A8C9,0x60E27BC08C6B47C4 +data8 0x19C367CDDCE8092A,0x8359C4768B961CA6 +data8 0xDDAF44D15719053E,0xA5FF07053F7E33E8 +data8 0x32C2DE4F98327DBB,0xC33D26EF6B1E5EF8 +data8 0x9F3A1F35CAF27F1D,0x87F121907C7C246A +data8 0xFA6ED5772D30433B,0x15C614B59D19C3C2 +data8 0xC4AD414D2C5D000C,0x467D862D71E39AC6 +data8 0x9B0062337CD2B497,0xA7B4D55537F63ED7 +data8 0x1810A3FC764D2A9D,0x64ABD770F87C6357 +data8 0xB07AE715175649C0,0xD9D63B3884A7CB23 +data8 0x24778AD623545AB9,0x1F001B0AF1DFCE19 +data8 0xFF319F6A1E666157,0x9947FBACD87F7EB7 +data8 0x652289E83260BFE6,0xCDC4EF09366CD43F +data8 0x5DD7DE16DE3B5892,0x9BDE2822D2E88628 +data8 0x4D58E232CAC616E3,0x08CB7DE050C017A7 +data8 0x1DF35BE01834132E,0x6212830148835B8E +data8 0xF57FB0ADF2E91E43,0x4A48D36710D8DDAA +data8 0x425FAECE616AA428,0x0AB499D3F2A6067F +data8 0x775C83C2A3883C61,0x78738A5A8CAFBDD7 +data8 0x6F63A62DCBBFF4EF,0x818D67C12645CA55 +data8 0x36D9CAD2A8288D61,0xC277C9121426049B +data8 0x4612C459C444C5C8,0x91B24DF31700AD43 +data8 0xD4E5492910D5FDFC,0xBE00CC941EEECE70 +data8 0xF53E1380F1ECC3E7,0xB328F8C79405933E +data8 0x71C1B3092EF3450B,0x9C12887B20AB9FB5 +data8 0x2EC292472F327B6D,0x550C90A7721FE76B +data8 0x96CB314A1679E279,0x4189DFF49794E884 +data8 0xE6E29731996BED88,0x365F5F0EFDBBB49A +data8 0x486CA46742727132,0x5D8DB8159F09E5BC +data8 0x25318D3974F71C05,0x30010C0D68084B58 +data8 0xEE2C90AA4702E774,0x24D6BDA67DF77248 +data8 0x6EEF169FA6948EF6,0x91B45153D1F20ACF +data8 0x3398207E4BF56863,0xB25F3EDD035D407F +data8 0x8985295255C06437,0x10D86D324832754C +data8 0x5BD4714E6E5445C1,0x090B69F52AD56614 +data8 0x9D072750045DDB3B,0xB4C576EA17F9877D +data8 0x6B49BA271D296996,0xACCCC65414AD6AE2 +data8 0x9089D98850722CBE,0xA4049407777030F3 +data8 0x27FC00A871EA49C2,0x663DE06483DD9797 +data8 0x3FA3FD94438C860D,0xDE41319D39928C70 +data8 0xDDE7B7173BDF082B,0x3715A0805C93805A +data8 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# |