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Diffstat (limited to 'sysdeps/ia64/fpu/e_powf.S')
| -rw-r--r-- | sysdeps/ia64/fpu/e_powf.S | 2065 |
1 files changed, 2065 insertions, 0 deletions
diff --git a/sysdeps/ia64/fpu/e_powf.S b/sysdeps/ia64/fpu/e_powf.S new file mode 100644 index 0000000000..d61bc79e5e --- /dev/null +++ b/sysdeps/ia64/fpu/e_powf.S @@ -0,0 +1,2065 @@ +.file "powf.s" + + +// Copyright (c) 2000 - 2005, 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 +// 02/03/00 Added p12 to definite over/under path. With odd power we did not +// maintain the sign of x in this path. +// 04/04/00 Unwind support added +// 04/19/00 pow(+-1,inf) now returns NaN +// pow(+-val, +-inf) returns 0 or inf, but now does not call error +// support +// Added s1 to fcvt.fx because invalid flag was incorrectly set. +// 08/15/00 Bundle added after call to __libm_error_support to properly +// set [the previously overwritten] GR_Parameter_RESULT. +// 09/07/00 Improved performance by eliminating bank conflicts and other stalls, +// and tweaking the critical path +// 09/08/00 Per c99, pow(+-1,inf) now returns 1, and pow(+1,nan) returns 1 +// 09/28/00 Updated NaN**0 path +// 01/20/01 Fixed denormal flag settings. +// 02/13/01 Improved speed. +// 03/19/01 Reordered exp polynomial to improve speed and eliminate monotonicity +// problem in round up, down, and to zero modes. Also corrected +// overflow result when x negative, y odd in round up, down, zero. +// 06/14/01 Added brace missing from bundle +// 12/10/01 Corrected case where x negative, 2^23 <= |y| < 2^24, y odd integer. +// 02/08/02 Fixed overflow/underflow cases that were not calling error support. +// 05/20/02 Cleaned up namespace and sf0 syntax +// 08/29/02 Improved Itanium 2 performance +// 02/10/03 Reordered header: .section, .global, .proc, .align +// 10/09/03 Modified algorithm to improve performance, reduce table size, and +// fix boundary case powf(2.0,-150.0) +// 03/31/05 Reformatted delimiters between data tables +// +// API +//============================================================== +// float powf(float x, float y) +// +// Overview of operation +//============================================================== +// +// Three steps... +// 1. Log(x) +// 2. y Log(x) +// 3. exp(y log(x)) +// +// This means we work with the absolute value of x and merge in the sign later. +// Log(x) = G + delta + r -rsq/2 + p +// G,delta depend on the exponent of x and table entries. The table entries are +// indexed by the exponent of x, called K. +// +// The G and delta come out of the reduction; r is the reduced x. +// +// B = frcpa(x) +// xB-1 is small means that B is the approximate inverse of x. +// +// Log(x) = Log( (1/B)(Bx) ) +// = Log(1/B) + Log(Bx) +// = Log(1/B) + Log( 1 + (Bx-1)) +// +// x = 2^K 1.x_1x_2.....x_52 +// B= frcpa(x) = 2^-k Cm +// Log(1/B) = Log(1/(2^-K Cm)) +// Log(1/B) = Log((2^K/ Cm)) +// Log(1/B) = K Log(2) + Log(1/Cm) +// +// Log(x) = K Log(2) + Log(1/Cm) + Log( 1 + (Bx-1)) +// +// If you take the significand of x, set the exponent to true 0, then Cm is +// the frcpa. We tabulate the Log(1/Cm) values. There are 256 of them. +// The frcpa table is indexed by 8 bits, the x_1 thru x_8. +// m = x_1x_2...x_8 is an 8-bit index. +// +// Log(1/Cm) = log(1/frcpa(1+m/256)) where m goes from 0 to 255. +// +// We tabluate as one double, T for single precision power +// +// Log(x) = (K Log(2)_hi + T) + (K Log(2)_lo) + Log( 1 + (Bx-1)) +// Log(x) = G + delta + Log( 1 + (Bx-1)) +// +// The Log( 1 + (Bx-1)) can be calculated as a series in r = Bx-1. +// +// Log( 1 + (Bx-1)) = r - rsq/2 + p +// where p = r^3(P0 + P1*r + P2*r^2) +// +// Then, +// +// yLog(x) = yG + y delta + y(r-rsq/2) + yp +// yLog(x) = Z1 + e3 + Z2 + Z3 +// +// +// exp(yLog(x)) = exp(Z1 + Z2) exp(Z3) exp(e3) +// +// +// exp(Z3) is another series. +// exp(e3) is approximated as f3 = 1 + e3 +// +// exp(Z1 + Z2) = exp(Z) +// Z (128/log2) = number of log2/128 in Z is N +// +// s = Z - N log2/128 +// +// exp(Z) = exp(s) exp(N log2/128) +// +// exp(r) = exp(Z - N log2/128) +// +// r = s + d = (Z - N (log2/128)_hi) -N (log2/128)_lo +// = Z - N (log2/128) +// +// Z = s+d +N (log2/128) +// +// exp(Z) = exp(s) (1+d) exp(N log2/128) +// +// N = M 128 + n +// +// N log2/128 = M log2 + n log2/128 +// +// n is 8 binary digits = n_7n_6...n_1 +// +// n log2/128 = n_7n_6n_5 16 log2/128 + n_4n_3n_2n_1 log2/128 +// n log2/128 = n_7n_6n_5 log2/8 + n_4n_3n_2n_1 log2/128 +// n log2/128 = I2 log2/8 + I1 log2/128 +// +// N log2/128 = M log2 + I2 log2/8 + I1 log2/128 +// +// exp(Z) = exp(s) (1+d) exp(log(2^M) + log(2^I2/8) + log(2^I1/128)) +// exp(Z) = exp(s) f12 (2^M) 2^I2/8 2^I1/128 +// +// I1, I2 are table indices. Use a series for exp(s). +// Then get exp(Z) +// +// exp(yLog(x)) = exp(Z) exp(Z3) f3 +// exp(yLog(x)) = exp(Z)f3 exp(Z3) +// exp(yLog(x)) = A exp(Z3) +// +// We actually calculate exp(Z3) -1. +// Then, +// exp(yLog(x)) = A + A( exp(Z3) -1) +// + +// Table Generation +//============================================================== + +// The log values +// ============== +// The operation (K*log2_hi) must be exact. K is the true exponent of x. +// If we allow gradual underflow (denormals), K can be represented in 12 bits +// (as a two's complement number). We assume 13 bits as an engineering +// precaution. +// +// +------------+----------------+-+ +// | 13 bits | 50 bits | | +// +------------+----------------+-+ +// 0 1 66 +// 2 34 +// +// So we want the lsb(log2_hi) to be 2^-50 +// We get log2 as a quad-extended (15-bit exponent, 128-bit significand) +// +// 0 fffe b17217f7d1cf79ab c9e3b39803f2f6af (4...) +// +// Consider numbering the bits left to right, starting at 0 thru 127. +// Bit 0 is the 2^-1 bit; bit 49 is the 2^-50 bit. +// +// ...79ab +// 0111 1001 1010 1011 +// 44 +// 89 +// +// So if we shift off the rightmost 14 bits, then (shift back only +// the top half) we get +// +// 0 fffe b17217f7d1cf4000 e6af278ece600fcb dabc000000000000 +// +// Put the right 64-bit signficand in an FR register, convert to double; +// it is exact. Put the next 128 bits into a quad register and round to double. +// The true exponent of the low part is -51. +// +// hi is 0 fffe b17217f7d1cf4000 +// lo is 0 ffcc e6af278ece601000 +// +// Convert to double memory format and get +// +// hi is 0x3fe62e42fefa39e8 +// lo is 0x3cccd5e4f1d9cc02 +// +// log2_hi + log2_lo is an accurate value for log2. +// +// +// The T and t values +// ================== +// A similar method is used to generate the T and t values. +// +// K * log2_hi + T must be exact. +// +// Smallest T,t +// ---------- +// The smallest T,t is +// T t +// 0x3f60040155d58800, 0x3c93bce0ce3ddd81 log(1/frcpa(1+0/256))= +1.95503e-003 +// +// The exponent is 0x3f6 (biased) or -9 (true). +// For the smallest T value, what we want is to clip the significand such that +// when it is shifted right by 9, its lsb is in the bit for 2^-51. The 9 is the +// specific for the first entry. In general, it is 0xffff - (biased 15-bit +// exponent). + +// Independently, what we have calculated is the table value as a quad +// precision number. +// Table entry 1 is +// 0 fff6 80200aaeac44ef38 338f77605fdf8000 +// +// We store this quad precision number in a data structure that is +// sign: 1 +// exponent: 15 +// signficand_hi: 64 (includes explicit bit) +// signficand_lo: 49 +// Because the explicit bit is included, the significand is 113 bits. +// +// Consider significand_hi for table entry 1. +// +// +// +-+--- ... -------+--------------------+ +// | | +// +-+--- ... -------+--------------------+ +// 0 1 4444444455555555556666 +// 2345678901234567890123 +// +// Labeled as above, bit 0 is 2^0, bit 1 is 2^-1, etc. +// Bit 42 is 2^-42. If we shift to the right by 9, the bit in +// bit 42 goes in 51. +// +// So what we want to do is shift bits 43 thru 63 into significand_lo. +// This is shifting bit 42 into bit 63, taking care to retain shifted-off bits. +// Then shifting (just with signficaand_hi) back into bit 42. +// +// The shift_value is 63-42 = 21. In general, this is +// 63 - (51 -(0xffff - 0xfff6)) +// For this example, it is +// 63 - (51 - 9) = 63 - 42 = 21 +// +// This means we are shifting 21 bits into significand_lo. We must maintain more +// that a 128-bit signficand not to lose bits. So before the shift we put the +// 128-bit significand into a 256-bit signficand and then shift. +// The 256-bit significand has four parts: hh, hl, lh, and ll. +// +// Start off with +// hh hl lh ll +// <64> <49><15_0> <64_0> <64_0> +// +// After shift by 21 (then return for significand_hi), +// <43><21_0> <21><43> <6><58_0> <64_0> +// +// Take the hh part and convert to a double. There is no rounding here. +// The conversion is exact. The true exponent of the high part is the same as +// the true exponent of the input quad. +// +// We have some 64 plus significand bits for the low part. In this example, we +// have 70 bits. We want to round this to a double. Put them in a quad and then +// do a quad fnorm. +// For this example the true exponent of the low part is +// true_exponent_of_high - 43 = true_exponent_of_high - (64-21) +// In general, this is +// true_exponent_of_high - (64 - shift_value) +// +// +// Largest T,t +// ---------- +// The largest T,t is +// 0x3fe62643fecf9742, 0x3c9e3147684bd37d log(1/frcpa(1+255/256))=+6.92171e-001 +// +// Table entry 256 is +// 0 fffe b1321ff67cba178c 51da12f4df5a0000 +// +// The shift value is +// 63 - (51 -(0xffff - 0xfffe)) = 13 +// +// The true exponent of the low part is +// true_exponent_of_high - (64 - shift_value) +// -1 - (64-13) = -52 +// Biased as a double, this is 0x3cb +// +// +// +// So then lsb(T) must be >= 2^-51 +// msb(Klog2_hi) <= 2^12 +// +// +--------+---------+ +// | 51 bits | <== largest T +// +--------+---------+ +// | 9 bits | 42 bits | <== smallest T +// +------------+----------------+-+ +// | 13 bits | 50 bits | | +// +------------+----------------+-+ +// +// Note: For powf only the table of T is needed + + +// Special Cases +//============================================================== + +// double float +// overflow error 24 30 + +// underflow error 25 31 + +// X zero Y zero +// +0 +0 +1 error 26 32 +// -0 +0 +1 error 26 32 +// +0 -0 +1 error 26 32 +// -0 -0 +1 error 26 32 + +// X zero Y negative +// +0 -odd integer +inf error 27 33 divide-by-zero +// -0 -odd integer -inf error 27 33 divide-by-zero +// +0 !-odd integer +inf error 27 33 divide-by-zero +// -0 !-odd integer +inf error 27 33 divide-by-zero +// +0 -inf +inf error 27 33 divide-by-zero +// -0 -inf +inf error 27 33 divide-by-zero + +// X zero Y positve +// +0 +odd integer +0 +// -0 +odd integer -0 +// +0 !+odd integer +0 +// -0 !+odd integer +0 +// +0 +inf +0 +// -0 +inf +0 +// +0 Y NaN quiet Y invalid if Y SNaN +// -0 Y NaN quiet Y invalid if Y SNaN + +// X one +// -1 Y inf +1 +// -1 Y NaN quiet Y invalid if Y SNaN +// +1 Y NaN +1 invalid if Y SNaN +// +1 Y any else +1 + +// X - Y not integer QNAN error 28 34 invalid + +// X NaN Y 0 +1 error 29 35 +// X NaN Y NaN quiet X invalid if X or Y SNaN +// X NaN Y any else quiet X invalid if X SNaN +// X !+1 Y NaN quiet Y invalid if Y SNaN + + +// X +inf Y >0 +inf +// X -inf Y >0, !odd integer +inf +// X -inf Y >0, odd integer -inf + +// X +inf Y <0 +0 +// X -inf Y <0, !odd integer +0 +// X -inf Y <0, odd integer -0 + +// X +inf Y =0 +1 +// X -inf Y =0 +1 + +// |X|<1 Y +inf +0 +// |X|<1 Y -inf +inf +// |X|>1 Y +inf +inf +// |X|>1 Y -inf +0 + +// X any Y =0 +1 + +// Assembly macros +//============================================================== + +// integer registers used + +pow_GR_exp_half = r10 +pow_GR_signexp_Xm1 = r11 +pow_GR_tmp = r11 + +pow_GR_signexp_X = r14 +pow_GR_17ones = r15 +pow_GR_Fpsr = r15 +pow_AD_P = r16 +pow_GR_rcs0_mask = r16 +pow_GR_exp_2tom8 = r17 +pow_GR_rcs0 = r17 +pow_GR_sig_X = r18 +pow_GR_10033 = r19 +pow_GR_16ones = r20 + +pow_AD_Tt = r21 +pow_GR_exp_X = r22 +pow_AD_Q = r23 +pow_GR_true_exp_X = r24 +pow_GR_y_zero = r25 + +pow_GR_exp_Y = r26 +pow_AD_tbl1 = r27 +pow_AD_tbl2 = r28 +pow_GR_offset = r29 +pow_GR_exp_Xm1 = r30 +pow_GR_xneg_yodd = r31 + +pow_GR_int_N = r38 +pow_GR_index1 = r39 +pow_GR_index2 = r40 + +pow_AD_T1 = r41 +pow_AD_T2 = r42 +pow_int_GR_M = r43 +pow_GR_sig_int_Y = r44 +pow_GR_sign_Y_Gpr = r45 + +pow_GR_17ones_m1 = r46 +pow_GR_one = r47 +pow_GR_sign_Y = r48 +pow_GR_signexp_Y_Gpr = r49 +pow_GR_exp_Y_Gpr = r50 + +pow_GR_true_exp_Y_Gpr = r51 +pow_GR_signexp_Y = r52 +pow_GR_x_one = r53 +pow_GR_big_pos = r55 + +pow_GR_big_neg = r56 + +GR_SAVE_B0 = r50 +GR_SAVE_GP = r51 +GR_SAVE_PFS = r52 + +GR_Parameter_X = r53 +GR_Parameter_Y = r54 +GR_Parameter_RESULT = r55 +pow_GR_tag = r56 + + +// floating point registers used + +POW_B = f32 +POW_NORM_X = f33 +POW_Xm1 = f34 +POW_r1 = f34 + +POW_NORM_Y = f37 +POW_Q2 = f38 +POW_eps = f39 +POW_P2 = f40 + +POW_P0 = f42 +POW_log2_lo = f43 +POW_r = f44 +POW_Q0_half = f45 + +POW_tmp = f47 +POW_log2_hi = f48 +POW_Q1 = f49 +POW_P1 = f50 + +POW_log2_by_128_hi = f51 +POW_inv_log2_by_128 = f52 +POW_rsq = f53 +POW_Yrcub = f54 +POW_log2_by_128_lo = f55 + +POW_xsq = f57 +POW_v2 = f59 +POW_T = f60 + +POW_RSHF = f62 +POW_v210 = f63 +POW_twoV = f65 + +POW_U = f66 +POW_G = f67 +POW_delta = f68 +POW_V = f70 + +POW_p = f71 +POW_Z = f72 +POW_e3 = f73 +POW_Z2 = f75 + +POW_W1 = f77 +POW_Z3 = f80 + +POW_Z3sq = f85 + +POW_Nfloat = f87 +POW_f3 = f89 +POW_q = f90 + +POW_T1 = f96 +POW_T2 = f97 +POW_2M = f98 +POW_s = f99 +POW_f12 = f100 + +POW_ssq = f101 +POW_T1T2 = f102 +POW_1ps = f103 +POW_A = f104 +POW_es = f105 + +POW_Xp1 = f106 +POW_int_K = f107 +POW_K = f108 +POW_f123 = f109 +POW_Gpr = f110 + +POW_Y_Gpr = f111 +POW_int_Y = f112 +POW_2Mqp1 = f113 + +POW_float_int_Y = f116 +POW_ftz_urm_f8 = f117 +POW_wre_urm_f8 = f118 +POW_big_neg = f119 +POW_big_pos = f120 + +// Data tables +//============================================================== + +RODATA + +.align 16 + +LOCAL_OBJECT_START(pow_table_P) +data8 0x80000000000018E5, 0x0000BFFD // P_1 +data8 0xb8aa3b295c17f0bc, 0x00004006 // inv_ln2_by_128 +// +// +data8 0x3FA5555555554A9E // Q_2 +data8 0x0000000000000000 // Pad +data8 0x3FC5555555554733 // Q_1 +data8 0x43e8000000000000 // Right shift constant for exp +data8 0xc9e3b39803f2f6af, 0x00003fb7 // ln2_by_128_lo +LOCAL_OBJECT_END(pow_table_P) + +LOCAL_OBJECT_START(pow_table_Q) +data8 0xCCCCCCCC4ED2BA7F, 0x00003FFC // P_2 +data8 0xAAAAAAAAAAAAB505, 0x00003FFD // P_0 +data8 0x3fe62e42fefa39e8, 0x3cccd5e4f1d9cc02 // log2 hi lo = +6.93147e-001 +data8 0xb17217f7d1cf79ab, 0x00003ff7 // ln2_by_128_hi +LOCAL_OBJECT_END(pow_table_Q) + + +LOCAL_OBJECT_START(pow_Tt) +data8 0x3f60040155d58800 // log(1/frcpa(1+0/256))= +1.95503e-003 +data8 0x3f78121214586a00 // log(1/frcpa(1+1/256))= +5.87661e-003 +data8 0x3f841929f9683200 // log(1/frcpa(1+2/256))= +9.81362e-003 +data8 0x3f8c317384c75f00 // log(1/frcpa(1+3/256))= +1.37662e-002 +data8 0x3f91a6b91ac73380 // log(1/frcpa(1+4/256))= +1.72376e-002 +data8 0x3f95ba9a5d9ac000 // log(1/frcpa(1+5/256))= +2.12196e-002 +data8 0x3f99d2a807432580 // log(1/frcpa(1+6/256))= +2.52177e-002 +data8 0x3f9d6b2725979800 // log(1/frcpa(1+7/256))= +2.87291e-002 +data8 0x3fa0c58fa19dfa80 // log(1/frcpa(1+8/256))= +3.27573e-002 +data8 0x3fa2954c78cbce00 // log(1/frcpa(1+9/256))= +3.62953e-002 +data8 0x3fa4a94d2da96c40 // log(1/frcpa(1+10/256))= +4.03542e-002 +data8 0x3fa67c94f2d4bb40 // log(1/frcpa(1+11/256))= +4.39192e-002 +data8 0x3fa85188b630f040 // log(1/frcpa(1+12/256))= +4.74971e-002 +data8 0x3faa6b8abe73af40 // log(1/frcpa(1+13/256))= +5.16017e-002 +data8 0x3fac441e06f72a80 // log(1/frcpa(1+14/256))= +5.52072e-002 +data8 0x3fae1e6713606d00 // log(1/frcpa(1+15/256))= +5.88257e-002 +data8 0x3faffa6911ab9300 // log(1/frcpa(1+16/256))= +6.24574e-002 +data8 0x3fb0ec139c5da600 // log(1/frcpa(1+17/256))= +6.61022e-002 +data8 0x3fb1dbd2643d1900 // log(1/frcpa(1+18/256))= +6.97605e-002 +data8 0x3fb2cc7284fe5f00 // log(1/frcpa(1+19/256))= +7.34321e-002 +data8 0x3fb3bdf5a7d1ee60 // log(1/frcpa(1+20/256))= +7.71173e-002 +data8 0x3fb4b05d7aa012e0 // log(1/frcpa(1+21/256))= +8.08161e-002 +data8 0x3fb580db7ceb5700 // log(1/frcpa(1+22/256))= +8.39975e-002 +data8 0x3fb674f089365a60 // log(1/frcpa(1+23/256))= +8.77219e-002 +data8 0x3fb769ef2c6b5680 // log(1/frcpa(1+24/256))= +9.14602e-002 +data8 0x3fb85fd927506a40 // log(1/frcpa(1+25/256))= +9.52125e-002 +data8 0x3fb9335e5d594980 // log(1/frcpa(1+26/256))= +9.84401e-002 +data8 0x3fba2b0220c8e5e0 // log(1/frcpa(1+27/256))= +1.02219e-001 +data8 0x3fbb0004ac1a86a0 // log(1/frcpa(1+28/256))= +1.05469e-001 +data8 0x3fbbf968769fca00 // log(1/frcpa(1+29/256))= +1.09274e-001 +data8 0x3fbccfedbfee13a0 // log(1/frcpa(1+30/256))= +1.12548e-001 +data8 0x3fbda727638446a0 // log(1/frcpa(1+31/256))= +1.15832e-001 +data8 0x3fbea3257fe10f60 // log(1/frcpa(1+32/256))= +1.19677e-001 +data8 0x3fbf7be9fedbfde0 // log(1/frcpa(1+33/256))= +1.22985e-001 +data8 0x3fc02ab352ff25f0 // log(1/frcpa(1+34/256))= +1.26303e-001 +data8 0x3fc097ce579d2040 // log(1/frcpa(1+35/256))= +1.29633e-001 +data8 0x3fc1178e8227e470 // log(1/frcpa(1+36/256))= +1.33531e-001 +data8 0x3fc185747dbecf30 // log(1/frcpa(1+37/256))= +1.36885e-001 +data8 0x3fc1f3b925f25d40 // log(1/frcpa(1+38/256))= +1.40250e-001 +data8 0x3fc2625d1e6ddf50 // log(1/frcpa(1+39/256))= +1.43627e-001 +data8 0x3fc2d1610c868130 // log(1/frcpa(1+40/256))= +1.47015e-001 +data8 0x3fc340c597411420 // log(1/frcpa(1+41/256))= +1.50414e-001 +data8 0x3fc3b08b6757f2a0 // log(1/frcpa(1+42/256))= +1.53825e-001 +data8 0x3fc40dfb08378000 // log(1/frcpa(1+43/256))= +1.56677e-001 +data8 0x3fc47e74e8ca5f70 // log(1/frcpa(1+44/256))= +1.60109e-001 +data8 0x3fc4ef51f6466de0 // log(1/frcpa(1+45/256))= +1.63553e-001 +data8 0x3fc56092e02ba510 // log(1/frcpa(1+46/256))= +1.67010e-001 +data8 0x3fc5d23857cd74d0 // log(1/frcpa(1+47/256))= +1.70478e-001 +data8 0x3fc6313a37335d70 // log(1/frcpa(1+48/256))= +1.73377e-001 +data8 0x3fc6a399dabbd380 // log(1/frcpa(1+49/256))= +1.76868e-001 +data8 0x3fc70337dd3ce410 // log(1/frcpa(1+50/256))= +1.79786e-001 +data8 0x3fc77654128f6120 // log(1/frcpa(1+51/256))= +1.83299e-001 +data8 0x3fc7e9d82a0b0220 // log(1/frcpa(1+52/256))= +1.86824e-001 +data8 0x3fc84a6b759f5120 // log(1/frcpa(1+53/256))= +1.89771e-001 +data8 0x3fc8ab47d5f5a300 // log(1/frcpa(1+54/256))= +1.92727e-001 +data8 0x3fc91fe490965810 // log(1/frcpa(1+55/256))= +1.96286e-001 +data8 0x3fc981634011aa70 // log(1/frcpa(1+56/256))= +1.99261e-001 +data8 0x3fc9f6c407089660 // log(1/frcpa(1+57/256))= +2.02843e-001 +data8 0x3fca58e729348f40 // log(1/frcpa(1+58/256))= +2.05838e-001 +data8 0x3fcabb55c31693a0 // log(1/frcpa(1+59/256))= +2.08842e-001 +data8 0x3fcb1e104919efd0 // log(1/frcpa(1+60/256))= +2.11855e-001 +data8 0x3fcb94ee93e367c0 // log(1/frcpa(1+61/256))= +2.15483e-001 +data8 0x3fcbf851c0675550 // log(1/frcpa(1+62/256))= +2.18516e-001 +data8 0x3fcc5c0254bf23a0 // log(1/frcpa(1+63/256))= +2.21558e-001 +data8 0x3fccc000c9db3c50 // log(1/frcpa(1+64/256))= +2.24609e-001 +data8 0x3fcd244d99c85670 // log(1/frcpa(1+65/256))= +2.27670e-001 +data8 0x3fcd88e93fb2f450 // log(1/frcpa(1+66/256))= +2.30741e-001 +data8 0x3fcdedd437eaef00 // log(1/frcpa(1+67/256))= +2.33820e-001 +data8 0x3fce530effe71010 // log(1/frcpa(1+68/256))= +2.36910e-001 +data8 0x3fceb89a1648b970 // log(1/frcpa(1+69/256))= +2.40009e-001 +data8 0x3fcf1e75fadf9bd0 // log(1/frcpa(1+70/256))= +2.43117e-001 +data8 0x3fcf84a32ead7c30 // log(1/frcpa(1+71/256))= +2.46235e-001 +data8 0x3fcfeb2233ea07c0 // log(1/frcpa(1+72/256))= +2.49363e-001 +data8 0x3fd028f9c7035c18 // log(1/frcpa(1+73/256))= +2.52501e-001 +data8 0x3fd05c8be0d96358 // log(1/frcpa(1+74/256))= +2.55649e-001 +data8 0x3fd085eb8f8ae790 // log(1/frcpa(1+75/256))= +2.58174e-001 +data8 0x3fd0b9c8e32d1910 // log(1/frcpa(1+76/256))= +2.61339e-001 +data8 0x3fd0edd060b78080 // log(1/frcpa(1+77/256))= +2.64515e-001 +data8 0x3fd122024cf00638 // log(1/frcpa(1+78/256))= +2.67701e-001 +data8 0x3fd14be2927aecd0 // log(1/frcpa(1+79/256))= +2.70257e-001 +data8 0x3fd180618ef18ad8 // log(1/frcpa(1+80/256))= +2.73461e-001 +data8 0x3fd1b50bbe2fc638 // log(1/frcpa(1+81/256))= +2.76675e-001 +data8 0x3fd1df4cc7cf2428 // log(1/frcpa(1+82/256))= +2.79254e-001 +data8 0x3fd214456d0eb8d0 // log(1/frcpa(1+83/256))= +2.82487e-001 +data8 0x3fd23ec5991eba48 // log(1/frcpa(1+84/256))= +2.85081e-001 +data8 0x3fd2740d9f870af8 // log(1/frcpa(1+85/256))= +2.88333e-001 +data8 0x3fd29ecdabcdfa00 // log(1/frcpa(1+86/256))= +2.90943e-001 +data8 0x3fd2d46602adcce8 // log(1/frcpa(1+87/256))= +2.94214e-001 +data8 0x3fd2ff66b04ea9d0 // log(1/frcpa(1+88/256))= +2.96838e-001 +data8 0x3fd335504b355a30 // log(1/frcpa(1+89/256))= +3.00129e-001 +data8 0x3fd360925ec44f58 // log(1/frcpa(1+90/256))= +3.02769e-001 +data8 0x3fd38bf1c3337e70 // log(1/frcpa(1+91/256))= +3.05417e-001 +data8 0x3fd3c25277333180 // log(1/frcpa(1+92/256))= +3.08735e-001 +data8 0x3fd3edf463c16838 // log(1/frcpa(1+93/256))= +3.11399e-001 +data8 0x3fd419b423d5e8c0 // log(1/frcpa(1+94/256))= +3.14069e-001 +data8 0x3fd44591e0539f48 // log(1/frcpa(1+95/256))= +3.16746e-001 +data8 0x3fd47c9175b6f0a8 // log(1/frcpa(1+96/256))= +3.20103e-001 +data8 0x3fd4a8b341552b08 // log(1/frcpa(1+97/256))= +3.22797e-001 +data8 0x3fd4d4f390890198 // log(1/frcpa(1+98/256))= +3.25498e-001 +data8 0x3fd501528da1f960 // log(1/frcpa(1+99/256))= +3.28206e-001 +data8 0x3fd52dd06347d4f0 // log(1/frcpa(1+100/256))= +3.30921e-001 +data8 0x3fd55a6d3c7b8a88 // log(1/frcpa(1+101/256))= +3.33644e-001 +data8 0x3fd5925d2b112a58 // log(1/frcpa(1+102/256))= +3.37058e-001 +data8 0x3fd5bf406b543db0 // log(1/frcpa(1+103/256))= +3.39798e-001 +data8 0x3fd5ec433d5c35a8 // log(1/frcpa(1+104/256))= +3.42545e-001 +data8 0x3fd61965cdb02c18 // log(1/frcpa(1+105/256))= +3.45300e-001 +data8 0x3fd646a84935b2a0 // log(1/frcpa(1+106/256))= +3.48063e-001 +data8 0x3fd6740add31de90 // log(1/frcpa(1+107/256))= +3.50833e-001 +data8 0x3fd6a18db74a58c0 // log(1/frcpa(1+108/256))= +3.53610e-001 +data8 0x3fd6cf31058670e8 // log(1/frcpa(1+109/256))= +3.56396e-001 +data8 0x3fd6f180e852f0b8 // log(1/frcpa(1+110/256))= +3.58490e-001 +data8 0x3fd71f5d71b894e8 // log(1/frcpa(1+111/256))= +3.61289e-001 +data8 0x3fd74d5aefd66d58 // log(1/frcpa(1+112/256))= +3.64096e-001 +data8 0x3fd77b79922bd378 // log(1/frcpa(1+113/256))= +3.66911e-001 +data8 0x3fd7a9b9889f19e0 // log(1/frcpa(1+114/256))= +3.69734e-001 +data8 0x3fd7d81b037eb6a0 // log(1/frcpa(1+115/256))= +3.72565e-001 +data8 0x3fd8069e33827230 // log(1/frcpa(1+116/256))= +3.75404e-001 +data8 0x3fd82996d3ef8bc8 // log(1/frcpa(1+117/256))= +3.77538e-001 +data8 0x3fd85855776dcbf8 // log(1/frcpa(1+118/256))= +3.80391e-001 +data8 0x3fd8873658327cc8 // log(1/frcpa(1+119/256))= +3.83253e-001 +data8 0x3fd8aa75973ab8c8 // log(1/frcpa(1+120/256))= +3.85404e-001 +data8 0x3fd8d992dc8824e0 // log(1/frcpa(1+121/256))= +3.88280e-001 +data8 0x3fd908d2ea7d9510 // log(1/frcpa(1+122/256))= +3.91164e-001 +data8 0x3fd92c59e79c0e50 // log(1/frcpa(1+123/256))= +3.93332e-001 +data8 0x3fd95bd750ee3ed0 // log(1/frcpa(1+124/256))= +3.96231e-001 +data8 0x3fd98b7811a3ee58 // log(1/frcpa(1+125/256))= +3.99138e-001 +data8 0x3fd9af47f33d4068 // log(1/frcpa(1+126/256))= +4.01323e-001 +data8 0x3fd9df270c1914a0 // log(1/frcpa(1+127/256))= +4.04245e-001 +data8 0x3fda0325ed14fda0 // log(1/frcpa(1+128/256))= +4.06442e-001 +data8 0x3fda33440224fa78 // log(1/frcpa(1+129/256))= +4.09379e-001 +data8 0x3fda57725e80c380 // log(1/frcpa(1+130/256))= +4.11587e-001 +data8 0x3fda87d0165dd198 // log(1/frcpa(1+131/256))= +4.14539e-001 +data8 0x3fdaac2e6c03f890 // log(1/frcpa(1+132/256))= +4.16759e-001 +data8 0x3fdadccc6fdf6a80 // log(1/frcpa(1+133/256))= +4.19726e-001 +data8 0x3fdb015b3eb1e790 // log(1/frcpa(1+134/256))= +4.21958e-001 +data8 0x3fdb323a3a635948 // log(1/frcpa(1+135/256))= +4.24941e-001 +data8 0x3fdb56fa04462908 // log(1/frcpa(1+136/256))= +4.27184e-001 +data8 0x3fdb881aa659bc90 // log(1/frcpa(1+137/256))= +4.30182e-001 +data8 0x3fdbad0bef3db160 // log(1/frcpa(1+138/256))= +4.32437e-001 +data8 0x3fdbd21297781c28 // log(1/frcpa(1+139/256))= +4.34697e-001 +data8 0x3fdc039236f08818 // log(1/frcpa(1+140/256))= +4.37718e-001 +data8 0x3fdc28cb1e4d32f8 // log(1/frcpa(1+141/256))= +4.39990e-001 +data8 0x3fdc4e19b84723c0 // log(1/frcpa(1+142/256))= +4.42267e-001 +data8 0x3fdc7ff9c74554c8 // log(1/frcpa(1+143/256))= +4.45311e-001 +data8 0x3fdca57b64e9db00 // log(1/frcpa(1+144/256))= +4.47600e-001 +data8 0x3fdccb130a5ceba8 // log(1/frcpa(1+145/256))= +4.49895e-001 +data8 0x3fdcf0c0d18f3268 // log(1/frcpa(1+146/256))= +4.52194e-001 +data8 0x3fdd232075b5a200 // log(1/frcpa(1+147/256))= +4.55269e-001 +data8 0x3fdd490246defa68 // log(1/frcpa(1+148/256))= +4.57581e-001 +data8 0x3fdd6efa918d25c8 // log(1/frcpa(1+149/256))= +4.59899e-001 +data8 0x3fdd9509707ae528 // log(1/frcpa(1+150/256))= +4.62221e-001 +data8 0x3fddbb2efe92c550 // log(1/frcpa(1+151/256))= +4.64550e-001 +data8 0x3fddee2f3445e4a8 // log(1/frcpa(1+152/256))= +4.67663e-001 +data8 0x3fde148a1a2726c8 // log(1/frcpa(1+153/256))= +4.70004e-001 +data8 0x3fde3afc0a49ff38 // log(1/frcpa(1+154/256))= +4.72350e-001 +data8 0x3fde6185206d5168 // log(1/frcpa(1+155/256))= +4.74702e-001 +data8 0x3fde882578823d50 // log(1/frcpa(1+156/256))= +4.77060e-001 +data8 0x3fdeaedd2eac9908 // log(1/frcpa(1+157/256))= +4.79423e-001 +data8 0x3fded5ac5f436be0 // log(1/frcpa(1+158/256))= +4.81792e-001 +data8 0x3fdefc9326d16ab8 // log(1/frcpa(1+159/256))= +4.84166e-001 +data8 0x3fdf2391a21575f8 // log(1/frcpa(1+160/256))= +4.86546e-001 +data8 0x3fdf4aa7ee031928 // log(1/frcpa(1+161/256))= +4.88932e-001 +data8 0x3fdf71d627c30bb0 // log(1/frcpa(1+162/256))= +4.91323e-001 +data8 0x3fdf991c6cb3b378 // log(1/frcpa(1+163/256))= +4.93720e-001 +data8 0x3fdfc07ada69a908 // log(1/frcpa(1+164/256))= +4.96123e-001 +data8 0x3fdfe7f18eb03d38 // log(1/frcpa(1+165/256))= +4.98532e-001 +data8 0x3fe007c053c5002c // log(1/frcpa(1+166/256))= +5.00946e-001 +data8 0x3fe01b942198a5a0 // log(1/frcpa(1+167/256))= +5.03367e-001 +data8 0x3fe02f74400c64e8 // log(1/frcpa(1+168/256))= +5.05793e-001 +data8 0x3fe04360be7603ac // log(1/frcpa(1+169/256))= +5.08225e-001 +data8 0x3fe05759ac47fe30 // log(1/frcpa(1+170/256))= +5.10663e-001 +data8 0x3fe06b5f1911cf50 // log(1/frcpa(1+171/256))= +5.13107e-001 +data8 0x3fe078bf0533c568 // log(1/frcpa(1+172/256))= +5.14740e-001 +data8 0x3fe08cd9687e7b0c // log(1/frcpa(1+173/256))= +5.17194e-001 +data8 0x3fe0a10074cf9018 // log(1/frcpa(1+174/256))= +5.19654e-001 +data8 0x3fe0b5343a234474 // log(1/frcpa(1+175/256))= +5.22120e-001 +data8 0x3fe0c974c89431cc // log(1/frcpa(1+176/256))= +5.24592e-001 +data8 0x3fe0ddc2305b9884 // log(1/frcpa(1+177/256))= +5.27070e-001 +data8 0x3fe0eb524bafc918 // log(1/frcpa(1+178/256))= +5.28726e-001 +data8 0x3fe0ffb54213a474 // log(1/frcpa(1+179/256))= +5.31214e-001 +data8 0x3fe114253da97d9c // log(1/frcpa(1+180/256))= +5.33709e-001 +data8 0x3fe128a24f1d9afc // log(1/frcpa(1+181/256))= +5.36210e-001 +data8 0x3fe1365252bf0864 // log(1/frcpa(1+182/256))= +5.37881e-001 +data8 0x3fe14ae558b4a92c // log(1/frcpa(1+183/256))= +5.40393e-001 +data8 0x3fe15f85a19c7658 // log(1/frcpa(1+184/256))= +5.42910e-001 +data8 0x3fe16d4d38c119f8 // log(1/frcpa(1+185/256))= +5.44592e-001 +data8 0x3fe18203c20dd130 // log(1/frcpa(1+186/256))= +5.47121e-001 +data8 0x3fe196c7bc4b1f38 // log(1/frcpa(1+187/256))= +5.49656e-001 +data8 0x3fe1a4a738b7a33c // log(1/frcpa(1+188/256))= +5.51349e-001 +data8 0x3fe1b981c0c9653c // log(1/frcpa(1+189/256))= +5.53895e-001 +data8 0x3fe1ce69e8bb1068 // log(1/frcpa(1+190/256))= +5.56447e-001 +data8 0x3fe1dc619de06944 // log(1/frcpa(1+191/256))= +5.58152e-001 +data8 0x3fe1f160a2ad0da0 // log(1/frcpa(1+192/256))= +5.60715e-001 +data8 0x3fe2066d7740737c // log(1/frcpa(1+193/256))= +5.63285e-001 +data8 0x3fe2147dba47a390 // log(1/frcpa(1+194/256))= +5.65001e-001 +data8 0x3fe229a1bc5ebac0 // log(1/frcpa(1+195/256))= +5.67582e-001 +data8 0x3fe237c1841a502c // log(1/frcpa(1+196/256))= +5.69306e-001 +data8 0x3fe24cfce6f80d98 // log(1/frcpa(1+197/256))= +5.71898e-001 +data8 0x3fe25b2c55cd5760 // log(1/frcpa(1+198/256))= +5.73630e-001 +data8 0x3fe2707f4d5f7c40 // log(1/frcpa(1+199/256))= +5.76233e-001 +data8 0x3fe285e0842ca380 // log(1/frcpa(1+200/256))= +5.78842e-001 +data8 0x3fe294294708b770 // log(1/frcpa(1+201/256))= +5.80586e-001 +data8 0x3fe2a9a2670aff0c // log(1/frcpa(1+202/256))= +5.83207e-001 +data8 0x3fe2b7fb2c8d1cc0 // log(1/frcpa(1+203/256))= +5.84959e-001 +data8 0x3fe2c65a6395f5f4 // log(1/frcpa(1+204/256))= +5.86713e-001 +data8 0x3fe2dbf557b0df40 // log(1/frcpa(1+205/256))= +5.89350e-001 +data8 0x3fe2ea64c3f97654 // log(1/frcpa(1+206/256))= +5.91113e-001 +data8 0x3fe3001823684d70 // log(1/frcpa(1+207/256))= +5.93762e-001 +data8 0x3fe30e97e9a8b5cc // log(1/frcpa(1+208/256))= +5.95531e-001 +data8 0x3fe32463ebdd34e8 // log(1/frcpa(1+209/256))= +5.98192e-001 +data8 0x3fe332f4314ad794 // log(1/frcpa(1+210/256))= +5.99970e-001 +data8 0x3fe348d90e7464cc // log(1/frcpa(1+211/256))= +6.02643e-001 +data8 0x3fe35779f8c43d6c // log(1/frcpa(1+212/256))= +6.04428e-001 +data8 0x3fe36621961a6a98 // log(1/frcpa(1+213/256))= +6.06217e-001 +data8 0x3fe37c299f3c3668 // log(1/frcpa(1+214/256))= +6.08907e-001 +data8 0x3fe38ae2171976e4 // log(1/frcpa(1+215/256))= +6.10704e-001 +data8 0x3fe399a157a603e4 // log(1/frcpa(1+216/256))= +6.12504e-001 +data8 0x3fe3afccfe77b9d0 // log(1/frcpa(1+217/256))= +6.15210e-001 +data8 0x3fe3be9d503533b4 // log(1/frcpa(1+218/256))= +6.17018e-001 +data8 0x3fe3cd7480b4a8a0 // log(1/frcpa(1+219/256))= +6.18830e-001 +data8 0x3fe3e3c43918f76c // log(1/frcpa(1+220/256))= +6.21554e-001 +data8 0x3fe3f2acb27ed6c4 // log(1/frcpa(1+221/256))= +6.23373e-001 +data8 0x3fe4019c2125ca90 // log(1/frcpa(1+222/256))= +6.25197e-001 +data8 0x3fe4181061389720 // log(1/frcpa(1+223/256))= +6.27937e-001 +data8 0x3fe42711518df544 // log(1/frcpa(1+224/256))= +6.29769e-001 +data8 0x3fe436194e12b6bc // log(1/frcpa(1+225/256))= +6.31604e-001 +data8 0x3fe445285d68ea68 // log(1/frcpa(1+226/256))= +6.33442e-001 +data8 0x3fe45bcc464c8938 // log(1/frcpa(1+227/256))= +6.36206e-001 +data8 0x3fe46aed21f117fc // log(1/frcpa(1+228/256))= +6.38053e-001 +data8 0x3fe47a1527e8a2d0 // log(1/frcpa(1+229/256))= +6.39903e-001 +data8 0x3fe489445efffcc8 // log(1/frcpa(1+230/256))= +6.41756e-001 +data8 0x3fe4a018bcb69834 // log(1/frcpa(1+231/256))= +6.44543e-001 +data8 0x3fe4af5a0c9d65d4 // log(1/frcpa(1+232/256))= +6.46405e-001 +data8 0x3fe4bea2a5bdbe84 // log(1/frcpa(1+233/256))= +6.48271e-001 +data8 0x3fe4cdf28f10ac44 // log(1/frcpa(1+234/256))= +6.50140e-001 +data8 0x3fe4dd49cf994058 // log(1/frcpa(1+235/256))= +6.52013e-001 +data8 0x3fe4eca86e64a680 // log(1/frcpa(1+236/256))= +6.53889e-001 +data8 0x3fe503c43cd8eb68 // log(1/frcpa(1+237/256))= +6.56710e-001 +data8 0x3fe513356667fc54 // log(1/frcpa(1+238/256))= +6.58595e-001 +data8 0x3fe522ae0738a3d4 // log(1/frcpa(1+239/256))= +6.60483e-001 +data8 0x3fe5322e26867854 // log(1/frcpa(1+240/256))= +6.62376e-001 +data8 0x3fe541b5cb979808 // log(1/frcpa(1+241/256))= +6.64271e-001 +data8 0x3fe55144fdbcbd60 // log(1/frcpa(1+242/256))= +6.66171e-001 +data8 0x3fe560dbc45153c4 // log(1/frcpa(1+243/256))= +6.68074e-001 +data8 0x3fe5707a26bb8c64 // log(1/frcpa(1+244/256))= +6.69980e-001 +data8 0x3fe587f60ed5b8fc // log(1/frcpa(1+245/256))= +6.72847e-001 +data8 0x3fe597a7977c8f30 // log(1/frcpa(1+246/256))= +6.74763e-001 +data8 0x3fe5a760d634bb88 // log(1/frcpa(1+247/256))= +6.76682e-001 +data8 0x3fe5b721d295f10c // log(1/frcpa(1+248/256))= +6.78605e-001 +data8 0x3fe5c6ea94431ef8 // log(1/frcpa(1+249/256))= +6.80532e-001 +data8 0x3fe5d6bb22ea86f4 // log(1/frcpa(1+250/256))= +6.82462e-001 +data8 0x3fe5e6938645d38c // log(1/frcpa(1+251/256))= +6.84397e-001 +data8 0x3fe5f673c61a2ed0 // log(1/frcpa(1+252/256))= +6.86335e-001 +data8 0x3fe6065bea385924 // log(1/frcpa(1+253/256))= +6.88276e-001 +data8 0x3fe6164bfa7cc068 // log(1/frcpa(1+254/256))= +6.90222e-001 +data8 0x3fe62643fecf9740 // log(1/frcpa(1+255/256))= +6.92171e-001 +LOCAL_OBJECT_END(pow_Tt) + + +// Table 1 is 2^(index_1/128) where +// index_1 goes from 0 to 15 +LOCAL_OBJECT_START(pow_tbl1) +data8 0x8000000000000000 , 0x00003FFF +data8 0x80B1ED4FD999AB6C , 0x00003FFF +data8 0x8164D1F3BC030773 , 0x00003FFF +data8 0x8218AF4373FC25EC , 0x00003FFF +data8 0x82CD8698AC2BA1D7 , 0x00003FFF +data8 0x8383594EEFB6EE37 , 0x00003FFF +data8 0x843A28C3ACDE4046 , 0x00003FFF +data8 0x84F1F656379C1A29 , 0x00003FFF +data8 0x85AAC367CC487B15 , 0x00003FFF +data8 0x8664915B923FBA04 , 0x00003FFF +data8 0x871F61969E8D1010 , 0x00003FFF +data8 0x87DB357FF698D792 , 0x00003FFF +data8 0x88980E8092DA8527 , 0x00003FFF +data8 0x8955EE03618E5FDD , 0x00003FFF +data8 0x8A14D575496EFD9A , 0x00003FFF +data8 0x8AD4C6452C728924 , 0x00003FFF +LOCAL_OBJECT_END(pow_tbl1) + + +// Table 2 is 2^(index_1/8) where +// index_2 goes from 0 to 7 +LOCAL_OBJECT_START(pow_tbl2) +data8 0x8000000000000000 , 0x00003FFF +data8 0x8B95C1E3EA8BD6E7 , 0x00003FFF +data8 0x9837F0518DB8A96F , 0x00003FFF +data8 0xA5FED6A9B15138EA , 0x00003FFF +data8 0xB504F333F9DE6484 , 0x00003FFF +data8 0xC5672A115506DADD , 0x00003FFF +data8 0xD744FCCAD69D6AF4 , 0x00003FFF +data8 0xEAC0C6E7DD24392F , 0x00003FFF +LOCAL_OBJECT_END(pow_tbl2) + +.section .text +GLOBAL_LIBM_ENTRY(powf) + +// Get exponent of x. Will be used to calculate K. +{ .mfi + getf.exp pow_GR_signexp_X = f8 + fms.s1 POW_Xm1 = f8,f1,f1 // Will be used for r1 if x>0 + mov pow_GR_17ones = 0x1FFFF +} +{ .mfi + addl pow_AD_P = @ltoff(pow_table_P), gp + fma.s1 POW_Xp1 = f8,f1,f1 // Will be used for r1 if x<0 + nop.i 999 +} +;; + +// Get significand of x. Will be used to get index to fetch T, Tt. +{ .mfi + getf.sig pow_GR_sig_X = f8 + frcpa.s1 POW_B, p6 = f1,f8 + mov pow_GR_exp_half = 0xFFFE // Exponent for 0.5 +} +{ .mfi + ld8 pow_AD_P = [pow_AD_P] + fma.s1 POW_NORM_X = f8,f1,f0 + mov pow_GR_exp_2tom8 = 0xFFF7 +} +;; + +// DOUBLE 0x10033 exponent limit at which y is an integer +{ .mfi + nop.m 999 + fcmp.lt.s1 p8,p9 = f8, f0 // Test for x<0 + addl pow_GR_10033 = 0x10033, r0 +} +{ .mfi + mov pow_GR_16ones = 0xFFFF + fma.s1 POW_NORM_Y = f9,f1,f0 + nop.i 999 +} +;; + +// p13 = TRUE ==> X is unorm +{ .mfi + setf.exp POW_Q0_half = pow_GR_exp_half // Form 0.5 + fclass.m p13,p0 = f8, 0x0b // Test for x unorm + adds pow_AD_Tt = pow_Tt - pow_table_P, pow_AD_P +} +{ .mfi + adds pow_AD_Q = pow_table_Q - pow_table_P, pow_AD_P + nop.f 999 + nop.i 999 +} +;; + +// p14 = TRUE ==> X is ZERO +{ .mfi + ldfe POW_P2 = [pow_AD_Q], 16 + fclass.m p14,p0 = f8, 0x07 + nop.i 999 +} +// Note POW_Xm1 and POW_r1 are used interchangably +{ .mfb + nop.m 999 +(p8) fnma.s1 POW_Xm1 = POW_Xp1,f1,f0 +(p13) br.cond.spnt POW_X_DENORM +} +;; + +// Continue normal and denormal paths here +POW_COMMON: +// p11 = TRUE ==> Y is a NAN +{ .mfi + and pow_GR_exp_X = pow_GR_signexp_X, pow_GR_17ones + fclass.m p11,p0 = f9, 0xc3 + nop.i 999 +} +{ .mfi + nop.m 999 + fms.s1 POW_r = POW_B, POW_NORM_X,f1 + mov pow_GR_y_zero = 0 +} +;; + +// Get exponent of |x|-1 to use in comparison to 2^-8 +{ .mmi + getf.exp pow_GR_signexp_Xm1 = POW_Xm1 + sub pow_GR_true_exp_X = pow_GR_exp_X, pow_GR_16ones + extr.u pow_GR_offset = pow_GR_sig_X, 55, 8 +} +;; + +{ .mfi + alloc r32=ar.pfs,2,19,4,0 + fcvt.fx.s1 POW_int_Y = POW_NORM_Y + shladd pow_AD_Tt = pow_GR_offset, 3, pow_AD_Tt +} +{ .mfi + setf.sig POW_int_K = pow_GR_true_exp_X + nop.f 999 + nop.i 999 +} +;; + +// p12 = TRUE if Y is ZERO +// Compute xsq to decide later if |x|=1 +{ .mfi + ldfe POW_P1 = [pow_AD_P], 16 + fclass.m p12,p0 = f9, 0x07 + nop.i 999 +} +{ .mfb + ldfe POW_P0 = [pow_AD_Q], 16 + fma.s1 POW_xsq = POW_NORM_X, POW_NORM_X, f0 +(p11) br.cond.spnt POW_Y_NAN // Branch if y=nan +} +;; + +{ .mmf + getf.exp pow_GR_signexp_Y = POW_NORM_Y + ldfd POW_T = [pow_AD_Tt] + fma.s1 POW_rsq = POW_r, POW_r,f0 +} +;; + +// p11 = TRUE ==> X is a NAN +{ .mfi + ldfpd POW_log2_hi, POW_log2_lo = [pow_AD_Q], 16 + fclass.m p11,p0 = POW_NORM_X, 0xc3 + nop.i 999 +} +{ .mfi + ldfe POW_inv_log2_by_128 = [pow_AD_P], 16 + fma.s1 POW_delta = f0,f0,f0 // delta=0 in case |x| near 1 +(p12) mov pow_GR_y_zero = 1 +} +;; + +{ .mfi + ldfd POW_Q2 = [pow_AD_P], 16 + fnma.s1 POW_twoV = POW_r, POW_Q0_half,f1 + and pow_GR_exp_Xm1 = pow_GR_signexp_Xm1, pow_GR_17ones +} +{ .mfi + nop.m 999 + fma.s1 POW_U = POW_NORM_Y,POW_r,f0 + nop.i 999 +} +;; + +// Determine if we will use the |x| near 1 path (p6) or normal path (p7) +{ .mfi + nop.m 999 + fcvt.xf POW_K = POW_int_K + cmp.lt p6,p7 = pow_GR_exp_Xm1, pow_GR_exp_2tom8 +} +{ .mfb + nop.m 999 + fma.s1 POW_G = f0,f0,f0 // G=0 in case |x| near 1 +(p11) br.cond.spnt POW_X_NAN // Branch if x=nan and y not nan +} +;; + +// If on the x near 1 path, assign r1 to r +{ .mfi + ldfpd POW_Q1, POW_RSHF = [pow_AD_P], 16 +(p6) fma.s1 POW_r = POW_r1, f1, f0 + nop.i 999 +} +{ .mfb + nop.m 999 +(p6) fma.s1 POW_rsq = POW_r1, POW_r1, f0 +(p14) br.cond.spnt POW_X_0 // Branch if x zero and y not nan +} +;; + +{ .mfi + getf.sig pow_GR_sig_int_Y = POW_int_Y +(p6) fnma.s1 POW_twoV = POW_r1, POW_Q0_half,f1 + and pow_GR_exp_Y = pow_GR_signexp_Y, pow_GR_17ones +} +{ .mfb + andcm pow_GR_sign_Y = pow_GR_signexp_Y, pow_GR_17ones +(p6) fma.s1 POW_U = POW_NORM_Y,POW_r1,f0 +(p12) br.cond.spnt POW_Y_0 // Branch if y=zero, x not zero or nan +} +;; + +{ .mfi + ldfe POW_log2_by_128_lo = [pow_AD_P], 16 +(p7) fma.s1 POW_Z2 = POW_twoV, POW_U, f0 + nop.i 999 +} +{ .mfi + ldfe POW_log2_by_128_hi = [pow_AD_Q], 16 + nop.f 999 + nop.i 999 +} +;; + +{ .mfi + nop.m 999 + fcvt.xf POW_float_int_Y = POW_int_Y + nop.i 999 +} +{ .mfi + nop.m 999 +(p7) fma.s1 POW_G = POW_K, POW_log2_hi, POW_T + adds pow_AD_tbl1 = pow_tbl1 - pow_Tt, pow_AD_Q +} +;; + +// p11 = TRUE ==> X is NEGATIVE but not inf +{ .mfi + nop.m 999 + fclass.m p11,p0 = POW_NORM_X, 0x1a + nop.i 999 +} +{ .mfi + nop.m 999 +(p7) fma.s1 POW_delta = POW_K, POW_log2_lo, f0 + adds pow_AD_tbl2 = pow_tbl2 - pow_tbl1, pow_AD_tbl1 +} +;; + +{ .mfi + nop.m 999 +(p6) fma.s1 POW_Z = POW_twoV, POW_U, f0 + nop.i 999 +} +{ .mfi + nop.m 999 + fma.s1 POW_v2 = POW_P1, POW_r, POW_P0 + nop.i 999 +} +;; + +// p11 = TRUE ==> X is NEGATIVE but not inf +// p12 = TRUE ==> X is NEGATIVE AND Y already even int +// p13 = TRUE ==> X is NEGATIVE AND Y possible int +{ .mfi + nop.m 999 +(p7) fma.s1 POW_Z = POW_NORM_Y, POW_G, POW_Z2 +(p11) cmp.gt.unc p12,p13 = pow_GR_exp_Y, pow_GR_10033 +} +{ .mfi + nop.m 999 + fma.s1 POW_Gpr = POW_G, f1, POW_r + nop.i 999 +} +;; + +{ .mfi + nop.m 999 + fma.s1 POW_Yrcub = POW_rsq, POW_U, f0 + nop.i 999 +} +{ .mfi + nop.m 999 + fma.s1 POW_p = POW_rsq, POW_P2, POW_v2 + nop.i 999 +} +;; + +// Test if x inf +{ .mfi + nop.m 999 + fclass.m p15,p0 = POW_NORM_X, 0x23 + nop.i 999 +} +// By adding RSHF (1.1000...*2^63) we put integer part in rightmost significand +{ .mfi + nop.m 999 + fma.s1 POW_W1 = POW_Z, POW_inv_log2_by_128, POW_RSHF + nop.i 999 +} +;; + +// p13 = TRUE ==> X is NEGATIVE AND Y possible int +// p10 = TRUE ==> X is NEG and Y is an int +// p12 = TRUE ==> X is NEG and Y is not an int +{ .mfi + nop.m 999 +(p13) fcmp.eq.unc.s1 p10,p12 = POW_float_int_Y, POW_NORM_Y + mov pow_GR_xneg_yodd = 0 +} +{ .mfi + nop.m 999 + fma.s1 POW_Y_Gpr = POW_NORM_Y, POW_Gpr, f0 + nop.i 999 +} +;; + +// p11 = TRUE ==> X is +1.0 +{ .mfi + nop.m 999 + fcmp.eq.s1 p11,p0 = POW_NORM_X, f1 + nop.i 999 +} +;; + +// Extract rounded integer from rightmost significand of POW_W1 +// By subtracting RSHF we get rounded integer POW_Nfloat +{ .mfi + getf.sig pow_GR_int_N = POW_W1 + fms.s1 POW_Nfloat = POW_W1, f1, POW_RSHF + nop.i 999 +} +{ .mfb + nop.m 999 + fma.s1 POW_Z3 = POW_p, POW_Yrcub, f0 +(p12) br.cond.spnt POW_X_NEG_Y_NONINT // Branch if x neg, y not integer +} +;; + +// p7 = TRUE ==> Y is +1.0 +// p12 = TRUE ==> X is NEGATIVE AND Y is an odd integer +{ .mfi + getf.exp pow_GR_signexp_Y_Gpr = POW_Y_Gpr + fcmp.eq.s1 p7,p0 = POW_NORM_Y, f1 // Test for y=1.0 +(p10) tbit.nz.unc p12,p0 = pow_GR_sig_int_Y,0 +} +{ .mfb + nop.m 999 +(p11) fma.s.s0 f8 = f1,f1,f0 // If x=1, result is +1 +(p15) br.cond.spnt POW_X_INF +} +;; + +// Test x and y and flag denormal +{ .mfi + nop.m 999 + fcmp.eq.s0 p15,p0 = f8,f9 + nop.i 999 +} +{ .mfb + nop.m 999 + fma.s1 POW_e3 = POW_NORM_Y, POW_delta, f0 +(p11) br.ret.spnt b0 // Early exit if x=1.0, result is +1 +} +;; + +{ .mfi +(p12) mov pow_GR_xneg_yodd = 1 + fnma.s1 POW_f12 = POW_Nfloat, POW_log2_by_128_lo, f1 + nop.i 999 +} +{ .mfb + nop.m 999 + fnma.s1 POW_s = POW_Nfloat, POW_log2_by_128_hi, POW_Z +(p7) br.ret.spnt b0 // Early exit if y=1.0, result is x +} +;; + +{ .mmi + and pow_GR_index1 = 0x0f, pow_GR_int_N + and pow_GR_index2 = 0x70, pow_GR_int_N + shr pow_int_GR_M = pow_GR_int_N, 7 // M = N/128 +} +;; + +{ .mfi + shladd pow_AD_T1 = pow_GR_index1, 4, pow_AD_tbl1 + fma.s1 POW_q = POW_Z3, POW_Q1, POW_Q0_half + add pow_int_GR_M = pow_GR_16ones, pow_int_GR_M +} +{ .mfi + add pow_AD_T2 = pow_AD_tbl2, pow_GR_index2 + fma.s1 POW_Z3sq = POW_Z3, POW_Z3, f0 + nop.i 999 +} +;; + +{ .mmi + ldfe POW_T1 = [pow_AD_T1] + ldfe POW_T2 = [pow_AD_T2] + nop.i 999 +} +;; + +// f123 = f12*(e3+1) = f12*e3+f12 +{ .mfi + setf.exp POW_2M = pow_int_GR_M + fma.s1 POW_f123 = POW_e3,POW_f12,POW_f12 + nop.i 999 +} +{ .mfi + nop.m 999 + fma.s1 POW_ssq = POW_s, POW_s, f0 + nop.i 999 +} +;; + +{ .mfi + nop.m 999 + fma.s1 POW_v2 = POW_s, POW_Q2, POW_Q1 + and pow_GR_exp_Y_Gpr = pow_GR_signexp_Y_Gpr, pow_GR_17ones +} +;; + +{ .mfi + cmp.ne p12,p13 = pow_GR_xneg_yodd, r0 + fma.s1 POW_q = POW_Z3sq, POW_q, POW_Z3 + sub pow_GR_true_exp_Y_Gpr = pow_GR_exp_Y_Gpr, pow_GR_16ones +} +;; + +// p8 TRUE ==> |Y(G + r)| >= 7 + +// single +// -2^7 -2^6 2^6 2^7 +// -----+-----+----+ ... +-----+-----+----- +// p8 | p9 | p8 +// | | p10 | | + +// Form signexp of constants to indicate overflow +{ .mfi + mov pow_GR_big_pos = 0x1007f + nop.f 999 + cmp.le p8,p9 = 7, pow_GR_true_exp_Y_Gpr +} +{ .mfi + mov pow_GR_big_neg = 0x3007f + nop.f 999 + andcm pow_GR_sign_Y_Gpr = pow_GR_signexp_Y_Gpr, pow_GR_17ones +} +;; + +// Form big positive and negative constants to test for possible overflow +// Scale both terms of the polynomial by POW_f123 +{ .mfi + setf.exp POW_big_pos = pow_GR_big_pos + fma.s1 POW_ssq = POW_ssq, POW_f123, f0 +(p9) cmp.le.unc p0,p10 = 6, pow_GR_true_exp_Y_Gpr +} +{ .mfb + setf.exp POW_big_neg = pow_GR_big_neg + fma.s1 POW_1ps = POW_s, POW_f123, POW_f123 +(p8) br.cond.spnt POW_OVER_UNDER_X_NOT_INF +} +;; + +{ .mfi + nop.m 999 +(p12) fnma.s1 POW_T1T2 = POW_T1, POW_T2, f0 + nop.i 999 +} +{ .mfi + nop.m 999 +(p13) fma.s1 POW_T1T2 = POW_T1, POW_T2, f0 + nop.i 999 +} +;; + +{ .mfi + nop.m 999 + fma.s1 POW_v210 = POW_s, POW_v2, POW_Q0_half + nop.i 999 +} +{ .mfi + nop.m 999 + fma.s1 POW_2Mqp1 = POW_2M, POW_q, POW_2M + nop.i 999 +} +;; + +{ .mfi + nop.m 999 + fma.s1 POW_es = POW_ssq, POW_v210, POW_1ps + nop.i 999 +} +{ .mfi + nop.m 999 + fma.s1 POW_A = POW_T1T2, POW_2Mqp1, f0 + nop.i 999 +} +;; + +// Dummy op to set inexact +{ .mfi + nop.m 999 + fma.s0 POW_tmp = POW_2M, POW_q, POW_2M + nop.i 999 +} +;; + +{ .mfb + nop.m 999 + fma.s.s0 f8 = POW_A, POW_es, f0 +(p10) br.ret.sptk b0 // Exit main branch if no over/underflow +} +;; + +// POSSIBLE_OVER_UNDER +// p6 = TRUE ==> Y_Gpr negative +// Result is already computed. We just need to know if over/underflow occurred. + +{ .mfb + cmp.eq p0,p6 = pow_GR_sign_Y_Gpr, r0 + nop.f 999 +(p6) br.cond.spnt POW_POSSIBLE_UNDER +} +;; + +// POSSIBLE_OVER +// We got an answer. +// overflow is a possibility, not a certainty + + +// We define an overflow when the answer with +// WRE set +// user-defined rounding mode + +// double +// Largest double is 7FE (biased double) +// 7FE - 3FF + FFFF = 103FE +// Create + largest_double_plus_ulp +// Create - largest_double_plus_ulp +// Calculate answer with WRE set. + +// single +// Largest single is FE (biased double) +// FE - 7F + FFFF = 1007E +// Create + largest_single_plus_ulp +// Create - largest_single_plus_ulp +// Calculate answer with WRE set. + +// Cases when answer is ldn+1 are as follows: +// ldn ldn+1 +// --+----------|----------+------------ +// | +// +inf +inf -inf +// RN RN +// RZ + +// Put in s2 (td set, wre set) +{ .mfi + nop.m 999 + fsetc.s2 0x7F,0x42 + nop.i 999 +} +;; + +{ .mfi + nop.m 999 + fma.s.s2 POW_wre_urm_f8 = POW_A, POW_es, f0 + nop.i 999 +} +;; + +// Return s2 to default +{ .mfi + nop.m 999 + fsetc.s2 0x7F,0x40 + nop.i 999 +} +;; + +// p7 = TRUE ==> yes, we have an overflow +{ .mfi + nop.m 999 + fcmp.ge.s1 p7, p8 = POW_wre_urm_f8, POW_big_pos + nop.i 999 +} +;; + +{ .mfi + nop.m 999 +(p8) fcmp.le.s1 p7, p0 = POW_wre_urm_f8, POW_big_neg + nop.i 999 +} +;; + +{ .mbb +(p7) mov pow_GR_tag = 30 +(p7) br.cond.spnt __libm_error_region // Branch if overflow + br.ret.sptk b0 // Exit if did not overflow +} +;; + + +POW_POSSIBLE_UNDER: +// We got an answer. input was < -2^9 but > -2^10 (double) +// We got an answer. input was < -2^6 but > -2^7 (float) +// underflow is a possibility, not a certainty + +// We define an underflow when the answer with +// ftz set +// is zero (tiny numbers become zero) +// Notice (from below) that if we have an unlimited exponent range, +// then there is an extra machine number E between the largest denormal and +// the smallest normal. +// So if with unbounded exponent we round to E or below, then we are +// tiny and underflow has occurred. +// But notice that you can be in a situation where we are tiny, namely +// rounded to E, but when the exponent is bounded we round to smallest +// normal. So the answer can be the smallest normal with underflow. +// E +// -----+--------------------+--------------------+----- +// | | | +// 1.1...10 2^-3fff 1.1...11 2^-3fff 1.0...00 2^-3ffe +// 0.1...11 2^-3ffe (biased, 1) +// largest dn smallest normal + +// Form small constant (2^-170) to correct underflow result near region of +// smallest denormal in round-nearest. + +// Put in s2 (td set, ftz set) +.pred.rel "mutex",p12,p13 +{ .mfi + mov pow_GR_Fpsr = ar40 // Read the fpsr--need to check rc.s0 + fsetc.s2 0x7F,0x41 + mov pow_GR_rcs0_mask = 0x0c00 // Set mask for rc.s0 +} +{ .mfi +(p12) mov pow_GR_tmp = 0x2ffff - 170 + nop.f 999 +(p13) mov pow_GR_tmp = 0x0ffff - 170 +} +;; + +{ .mfi + setf.exp POW_eps = pow_GR_tmp // Form 2^-170 + fma.s.s2 POW_ftz_urm_f8 = POW_A, POW_es, f0 + nop.i 999 +} +;; + +// Return s2 to default +{ .mfi + nop.m 999 + fsetc.s2 0x7F,0x40 + nop.i 999 +} +;; + +// p7 = TRUE ==> yes, we have an underflow +{ .mfi + nop.m 999 + fcmp.eq.s1 p7, p0 = POW_ftz_urm_f8, f0 + nop.i 999 +} +;; + +{ .mmi +(p7) and pow_GR_rcs0 = pow_GR_rcs0_mask, pow_GR_Fpsr // Isolate rc.s0 +;; +(p7) cmp.eq.unc p6,p0 = pow_GR_rcs0, r0 // Test for round to nearest + nop.i 999 +} +;; + +// Tweak result slightly if underflow to get correct rounding near smallest +// denormal if round-nearest +{ .mfi + nop.m 999 +(p6) fms.s.s0 f8 = POW_A, POW_es, POW_eps + nop.i 999 +} +{ .mbb +(p7) mov pow_GR_tag = 31 +(p7) br.cond.spnt __libm_error_region // Branch if underflow + br.ret.sptk b0 // Exit if did not underflow +} +;; + +POW_X_DENORM: +// Here if x unorm. Use the NORM_X for getf instructions, and then back +// to normal path +{ .mfi + getf.exp pow_GR_signexp_X = POW_NORM_X + nop.f 999 + nop.i 999 +} +;; + +{ .mib + getf.sig pow_GR_sig_X = POW_NORM_X + nop.i 999 + br.cond.sptk POW_COMMON +} +;; + +POW_X_0: +// Here if x=0 and y not nan +// +// We have the following cases: +// p6 x=0 and y>0 and is an integer (may be even or odd) +// p7 x=0 and y>0 and is NOT an integer, return +0 +// p8 x=0 and y>0 and so big as to always be an even integer, return +0 +// p9 x=0 and y>0 and may not be integer +// p10 x=0 and y>0 and is an odd integer, return x +// p11 x=0 and y>0 and is an even integer, return +0 +// p12 used in dummy fcmp to set denormal flag if y=unorm +// p13 x=0 and y>0 +// p14 x=0 and y=0, branch to code for calling error handling +// p15 x=0 and y<0, branch to code for calling error handling +// +{ .mfi + getf.sig pow_GR_sig_int_Y = POW_int_Y // Get signif of int_Y + fcmp.lt.s1 p15,p13 = f9, f0 // Test for y<0 + and pow_GR_exp_Y = pow_GR_signexp_Y, pow_GR_17ones +} +{ .mfb + cmp.ne p14,p0 = pow_GR_y_zero,r0 // Test for y=0 + fcvt.xf POW_float_int_Y = POW_int_Y +(p14) br.cond.spnt POW_X_0_Y_0 // Branch if x=0 and y=0 +} +;; + +// If x=0 and y>0, test y and flag denormal +{ .mfb +(p13) cmp.gt.unc p8,p9 = pow_GR_exp_Y, pow_GR_10033 // Test y +big = even int +(p13) fcmp.eq.s0 p12,p0 = f9,f0 // If x=0, y>0 dummy op to flag denormal +(p15) br.cond.spnt POW_X_0_Y_NEG // Branch if x=0 and y<0 +} +;; + +// Here if x=0 and y>0 +{ .mfi + nop.m 999 +(p9) fcmp.eq.unc.s1 p6,p7 = POW_float_int_Y, POW_NORM_Y // Test y=int + nop.i 999 +} +{ .mfi + nop.m 999 +(p8) fma.s.s0 f8 = f0,f0,f0 // If x=0, y>0 and large even int, return +0 + nop.i 999 +} +;; + +{ .mfi + nop.m 999 +(p7) fma.s.s0 f8 = f0,f0,f0 // Result +0 if x=0 and y>0 and not integer +(p6) tbit.nz.unc p10,p11 = pow_GR_sig_int_Y,0 // If y>0 int, test y even/odd +} +;; + +// Note if x=0, y>0 and odd integer, just return x +{ .mfb + nop.m 999 +(p11) fma.s.s0 f8 = f0,f0,f0 // Result +0 if x=0 and y even integer + br.ret.sptk b0 // Exit if x=0 and y>0 +} +;; + +POW_X_0_Y_0: +// When X is +-0 and Y is +-0, IEEE returns 1.0 +// We call error support with this value + +{ .mfb + mov pow_GR_tag = 32 + fma.s.s0 f8 = f1,f1,f0 + br.cond.sptk __libm_error_region +} +;; + +POW_X_0_Y_NEG: +// When X is +-0 and Y is negative, IEEE returns +// X Y answer +// +0 -odd int +inf +// -0 -odd int -inf + +// +0 !-odd int +inf +// -0 !-odd int +inf + +// p6 == Y is a floating point number outside the integer. +// Hence it is an integer and is even. +// return +inf + +// p7 == Y is a floating point number within the integer range. +// p9 == (int_Y = NORM_Y), Y is an integer, which may be odd or even. +// p11 odd +// return (sign_of_x)inf +// p12 even +// return +inf +// p10 == Y is not an integer +// return +inf +// + +{ .mfi + nop.m 999 + nop.f 999 + cmp.gt p6,p7 = pow_GR_exp_Y, pow_GR_10033 +} +;; + +{ .mfi + mov pow_GR_tag = 33 +(p7) fcmp.eq.unc.s1 p9,p10 = POW_float_int_Y, POW_NORM_Y + nop.i 999 +} +;; + +{ .mfb + nop.m 999 +(p6) frcpa.s0 f8,p13 = f1, f0 +(p6) br.cond.sptk __libm_error_region // x=0, y<0, y large neg int +} +;; + +{ .mfb + nop.m 999 +(p10) frcpa.s0 f8,p13 = f1, f0 +(p10) br.cond.sptk __libm_error_region // x=0, y<0, y not int +} +;; + +// x=0, y<0, y an int +{ .mib + nop.m 999 +(p9) tbit.nz.unc p11,p12 = pow_GR_sig_int_Y,0 + nop.b 999 +} +;; + +{ .mfi + nop.m 999 +(p12) frcpa.s0 f8,p13 = f1,f0 + nop.i 999 +} +;; + +{ .mfb + nop.m 999 +(p11) frcpa.s0 f8,p13 = f1,f8 + br.cond.sptk __libm_error_region +} +;; + + +POW_Y_0: +// Here for y zero, x anything but zero and nan +// Set flag if x denormal +// Result is +1.0 +{ .mfi + nop.m 999 + fcmp.eq.s0 p6,p0 = f8,f0 // Sets flag if x denormal + nop.i 999 +} +{ .mfb + nop.m 999 + fma.s.s0 f8 = f1,f1,f0 + br.ret.sptk b0 +} +;; + + +POW_X_INF: +// Here when X is +-inf + +// X +inf Y +inf +inf +// X -inf Y +inf +inf + +// X +inf Y >0 +inf +// X -inf Y >0, !odd integer +inf <== (-inf)^0.5 = +inf !! +// X -inf Y >0, odd integer -inf + +// X +inf Y -inf +0 +// X -inf Y -inf +0 + +// X +inf Y <0 +0 +// X -inf Y <0, !odd integer +0 +// X -inf Y <0, odd integer -0 + +// X + inf Y=+0 +1 +// X + inf Y=-0 +1 +// X - inf Y=+0 +1 +// X - inf Y=-0 +1 + +// p13 == Y negative +// p14 == Y positive + +// p6 == Y is a floating point number outside the integer. +// Hence it is an integer and is even. +// p13 == (Y negative) +// return +inf +// p14 == (Y positive) +// return +0 + +// p7 == Y is a floating point number within the integer range. +// p9 == (int_Y = NORM_Y), Y is an integer, which may be odd or even. +// p11 odd +// p13 == (Y negative) +// return (sign_of_x)inf +// p14 == (Y positive) +// return (sign_of_x)0 +// pxx even +// p13 == (Y negative) +// return +inf +// p14 == (Y positive) +// return +0 + +// pxx == Y is not an integer +// p13 == (Y negative) +// return +inf +// p14 == (Y positive) +// return +0 +// + +// If x=inf, test y and flag denormal +{ .mfi + nop.m 999 + fcmp.eq.s0 p10,p11 = f9,f0 + nop.i 999 +} +;; + +{ .mfi + nop.m 999 + fcmp.lt.s0 p13,p14 = POW_NORM_Y,f0 + cmp.gt p6,p7 = pow_GR_exp_Y, pow_GR_10033 +} +{ .mfi + nop.m 999 + fclass.m p12,p0 = f9, 0x23 //@inf + nop.i 999 +} +;; + +{ .mfi + nop.m 999 + fclass.m p15,p0 = f9, 0x07 //@zero + nop.i 999 +} +;; + +{ .mfb + nop.m 999 +(p15) fmerge.s f8 = f1,f1 // Return +1.0 if x=inf, y=0 +(p15) br.ret.spnt b0 // Exit if x=inf, y=0 +} +;; + +{ .mfi + nop.m 999 +(p14) frcpa.s1 f8,p10 = f1,f0 // If x=inf, y>0, assume result +inf + nop.i 999 +} +{ .mfb + nop.m 999 +(p13) fma.s.s0 f8 = f0,f0,f0 // If x=inf, y<0, assume result +0.0 +(p12) br.ret.spnt b0 // Exit if x=inf, y=inf +} +;; + +// Here if x=inf, and 0 < |y| < inf. Need to correct results if y odd integer. +{ .mfi + nop.m 999 +(p7) fcmp.eq.unc.s1 p9,p0 = POW_float_int_Y, POW_NORM_Y // Is y integer? + nop.i 999 +} +;; + +{ .mfi + nop.m 999 + nop.f 999 +(p9) tbit.nz.unc p11,p0 = pow_GR_sig_int_Y,0 // Test for y odd integer +} +;; + +{ .mfb + nop.m 999 +(p11) fmerge.s f8 = POW_NORM_X,f8 // If y odd integer use sign of x + br.ret.sptk b0 // Exit for x=inf, 0 < |y| < inf +} +;; + + +POW_X_NEG_Y_NONINT: +// When X is negative and Y is a non-integer, IEEE +// returns a qnan indefinite. +// We call error support with this value + +{ .mfb + mov pow_GR_tag = 34 + frcpa.s0 f8,p6 = f0,f0 + br.cond.sptk __libm_error_region +} +;; + +POW_X_NAN: +// Here if x=nan, y not nan +{ .mfi + nop.m 999 + fclass.m p9,p13 = f9, 0x07 // Test y=zero + nop.i 999 +} +;; + +{ .mfb + nop.m 999 +(p13) fma.s.s0 f8 = f8,f1,f0 +(p13) br.ret.sptk b0 // Exit if x nan, y anything but zero or nan +} +;; + +POW_X_NAN_Y_0: +// When X is a NAN and Y is zero, IEEE returns 1. +// We call error support with this value. +{ .mfi + nop.m 999 + fcmp.eq.s0 p6,p0 = f8,f0 // Dummy op to set invalid on snan + nop.i 999 +} +{ .mfb + mov pow_GR_tag = 35 + fma.s.s0 f8 = f0,f0,f1 + br.cond.sptk __libm_error_region +} +;; + + +POW_OVER_UNDER_X_NOT_INF: + +// p8 is TRUE for overflow +// p9 is TRUE for underflow + +// if y is infinity, we should not over/underflow + +{ .mfi + nop.m 999 + fcmp.eq.s1 p14, p13 = POW_xsq,f1 // Test |x|=1 + cmp.eq p8,p9 = pow_GR_sign_Y_Gpr, r0 +} +;; + +{ .mfi + nop.m 999 +(p14) fclass.m.unc p15, p0 = f9, 0x23 // If |x|=1, test y=inf + nop.i 999 +} +{ .mfi + nop.m 999 +(p13) fclass.m.unc p11,p0 = f9, 0x23 // If |x| not 1, test y=inf + nop.i 999 +} +;; + +// p15 = TRUE if |x|=1, y=inf, return +1 +{ .mfb + nop.m 999 +(p15) fma.s.s0 f8 = f1,f1,f0 // If |x|=1, y=inf, result +1 +(p15) br.ret.spnt b0 // Exit if |x|=1, y=inf +} +;; + +.pred.rel "mutex",p8,p9 +{ .mfb +(p8) setf.exp f8 = pow_GR_17ones // If exp(+big), result inf +(p9) fmerge.s f8 = f0,f0 // If exp(-big), result 0 +(p11) br.ret.sptk b0 // Exit if |x| not 1, y=inf +} +;; + +{ .mfb + nop.m 999 + nop.f 999 + br.cond.sptk POW_OVER_UNDER_ERROR // Branch if y not inf +} +;; + + +POW_Y_NAN: +// Here if y=nan, x anything +// If x = +1 then result is +1, else result is quiet Y +{ .mfi + nop.m 999 + fcmp.eq.s1 p10,p9 = POW_NORM_X, f1 + nop.i 999 +} +;; + +{ .mfi + nop.m 999 +(p10) fcmp.eq.s0 p6,p0 = f9,f1 // Set invalid, even if x=+1 + nop.i 999 +} +;; + +{ .mfi + nop.m 999 +(p10) fma.s.s0 f8 = f1,f1,f0 + nop.i 999 +} +{ .mfb + nop.m 999 +(p9) fma.s.s0 f8 = f9,f8,f0 + br.ret.sptk b0 // Exit y=nan +} +;; + + +POW_OVER_UNDER_ERROR: +// Here if we have overflow or underflow. +// Enter with p12 true if x negative and y odd int to force -0 or -inf + +{ .mfi + sub pow_GR_17ones_m1 = pow_GR_17ones, r0, 1 + nop.f 999 + mov pow_GR_one = 0x1 +} +;; + +// overflow, force inf with O flag +{ .mmb +(p8) mov pow_GR_tag = 30 +(p8) setf.exp POW_tmp = pow_GR_17ones_m1 + nop.b 999 +} +;; + +// underflow, force zero with I, U flags +{ .mmi +(p9) mov pow_GR_tag = 31 +(p9) setf.exp POW_tmp = pow_GR_one + nop.i 999 +} +;; + +{ .mfi + nop.m 999 + fma.s.s0 f8 = POW_tmp, POW_tmp, f0 + nop.i 999 +} +;; + +// p12 x is negative and y is an odd integer, change sign of result +{ .mfi + nop.m 999 +(p12) fnma.s.s0 f8 = POW_tmp, POW_tmp, f0 + nop.i 999 +} +;; + +GLOBAL_LIBM_END(powf) + + +LOCAL_LIBM_ENTRY(__libm_error_region) + +.prologue +{ .mfi + add GR_Parameter_Y=-32,sp // Parameter 2 value + nop.f 0 +.save ar.pfs,GR_SAVE_PFS + mov GR_SAVE_PFS=ar.pfs // Save ar.pfs +} +{ .mfi +.fframe 64 + add sp=-64,sp // Create new stack + nop.f 0 + mov GR_SAVE_GP=gp // Save gp +};; + +{ .mmi + stfs [GR_Parameter_Y] = POW_NORM_Y,16 // STORE Parameter 2 on stack + add GR_Parameter_X = 16,sp // Parameter 1 address +.save b0, GR_SAVE_B0 + mov GR_SAVE_B0=b0 // Save b0 +};; + +.body +{ .mib + stfs [GR_Parameter_X] = POW_NORM_X // STORE Parameter 1 on stack + add GR_Parameter_RESULT = 0,GR_Parameter_Y // Parameter 3 address + nop.b 0 +} +{ .mib + stfs [GR_Parameter_Y] = f8 // STORE Parameter 3 on stack + add GR_Parameter_Y = -16,GR_Parameter_Y + br.call.sptk b0=__libm_error_support# // Call error handling function +};; + +{ .mmi + add GR_Parameter_RESULT = 48,sp + nop.m 0 + nop.i 0 +};; + +{ .mmi + ldfs f8 = [GR_Parameter_RESULT] // Get return result off stack +.restore sp + add sp = 64,sp // Restore stack pointer + mov b0 = GR_SAVE_B0 // Restore return address +};; + +{ .mib + mov gp = GR_SAVE_GP // Restore gp + mov ar.pfs = GR_SAVE_PFS // Restore ar.pfs + br.ret.sptk b0 // Return +};; + +LOCAL_LIBM_END(__libm_error_region) + +.type __libm_error_support#,@function +.global __libm_error_support# |