.file "pow.s"
// Copyright (c) 2000 - 2005, Intel Corporation
// All rights reserved.
//
// Contributed 2000 by the Intel Numerics Group, Intel Corporation
//
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// modification, are permitted provided that the following conditions are
// met:
//
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//
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// notice, this list of conditions and the following disclaimer in the
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//
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//
// 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, 2^52 <= |y| < 2^53, y odd integer.
// 12/20/01 Fixed monotonity problem in round to nearest.
// 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
// 09/21/02 Added branch for |y*log(x)|<2^-11 to fix monotonicity problems.
// 02/10/03 Reordered header: .section, .global, .proc, .align
// 03/31/05 Reformatted delimiters between data tables
//
// API
//==============================================================
// double pow(double x, double 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 two doubles, T and t, where T +t is the value itself.
//
// Log(x) = (K Log(2)_hi + T) + (Log(2)_hi + t) + 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
//
// Then,
//
// yLog(x) = yG + y delta + y(r-rsq/2) + yp
// yLog(x) = Z1 + e3 + Z2 + Z3 + (e2 + e3)
//
//
// exp(yLog(x)) = exp(Z1 + Z2 + Z3) exp(e1 + e2 + e3)
//
//
// exp(Z3) is another series.
// exp(e1 + e2 + e3) is approximated as f3 = 1 + (e1 + e2 + e3)
//
// Z1 (128/log2) = number of log2/128 in Z1 is N1
// Z2 (128/log2) = number of log2/128 in Z2 is N2
//
// s1 = Z1 - N1 log2/128
// s2 = Z2 - N2 log2/128
//
// s = s1 + s2
// N = N1 + N2
//
// exp(Z1 + Z2) = exp(Z)
// 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) (1+d1) (1+d2)(2^M) 2^I2/8 2^I1/128
// exp(Z) = exp(s) f1 f2 (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(Z1 + Z2 + Z3) exp(e1 + e2 + e3)
// 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 | |
// +------------+----------------+-+
// 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_signexp_X = r14
pow_GR_17ones = r15
pow_AD_P = r16
pow_GR_exp_2tom8 = 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_signexp_Xm1 = r35
pow_GR_int_W1 = r36
pow_GR_int_W2 = r37
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_exp_2toM63 = r54
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_P4 = f35
POW_P5 = f36
POW_NORM_Y = f37
POW_Q2 = f38
POW_Q3 = f39
POW_P2 = f40
POW_P3 = f41
POW_P0 = f42
POW_log2_lo = f43
POW_r = f44
POW_Q0_half = f45
POW_Q1 = f46
POW_tmp = f47
POW_log2_hi = f48
POW_Q4 = 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_v6 = f56
POW_xsq = f57
POW_v4 = f58
POW_v2 = f59
POW_T = f60
POW_Tt = f61
POW_RSHF = f62
POW_v21ps = f63
POW_s4 = f64
POW_twoV = f65
POW_U = f66
POW_G = f67
POW_delta = f68
POW_v3 = f69
POW_V = f70
POW_p = f71
POW_Z1 = f72
POW_e3 = f73
POW_e2 = f74
POW_Z2 = f75
POW_e1 = f76
POW_W1 = f77
POW_UmZ2 = f78
POW_W2 = f79
POW_Z3 = f80
POW_int_W1 = f81
POW_e12 = f82
POW_int_W2 = f83
POW_UmZ2pV = f84
POW_Z3sq = f85
POW_e123 = f86
POW_N1float = f87
POW_N2float = f88
POW_f3 = f89
POW_q = f90
POW_s1 = f91
POW_Nfloat = f92
POW_s2 = f93
POW_f2 = f94
POW_f1 = f95
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_abs_q = f114
POW_2toM63 = f115
POW_float_int_Y = f116
POW_ftz_urm_f8 = f117
POW_wre_urm_f8 = f118
POW_big_neg = f119
POW_big_pos = f120
POW_GY_Z2 = f121
POW_pYrcub_e3 = f122
POW_d = f123
POW_d2 = f124
POW_poly_d_hi = f121
POW_poly_d_lo = f122
POW_poly_d = f121
// Data tables
//==============================================================
RODATA
.align 16
LOCAL_OBJECT_START(pow_table_P)
data8 0x8000F7B249FF332D, 0x0000BFFC // P_5
data8 0xAAAAAAA9E7902C7F, 0x0000BFFC // P_3
data8 0x80000000000018E5, 0x0000BFFD // P_1
data8 0xb8aa3b295c17f0bc, 0x00004006 // inv_ln2_by_128
//
//
data8 0x3FA5555555554A9E // Q_2
data8 0x3F8111124F4DD9F9 // Q_3
data8 0x3FE0000000000000 // Q_0
data8 0x3FC5555555554733 // Q_1
data8 0x3F56C16D9360FFA0 // Q_4
data8 0x43e8000000000000 // Right shift constant for exp
data8 0xc9e3b39803f2f6af, 0x00003fb7 // ln2_by_128_lo
data8 0x0000000000000000 // pad to eliminate bank conflicts with pow_table_Q
data8 0x0000000000000000 // pad to eliminate bank conflicts with pow_table_Q
LOCAL_OBJECT_END(pow_table_P)
LOCAL_OBJECT_START(pow_table_Q)
data8 0x9249FE7F0DC423CF, 0x00003FFC // P_4
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, 0x3c93bce0ce3ddd81 // log(1/frcpa(1+0/256))= +1.95503e-003
data8 0x3f78121214586a00, 0x3cb540e0a5cfc9bc // log(1/frcpa(1+1/256))= +5.87661e-003
data8 0x3f841929f9683200, 0x3cbdf1d57404da1f // log(1/frcpa(1+2/256))= +9.81362e-003
data8 0x3f8c317384c75f00, 0x3c69806208c04c22 // log(1/frcpa(1+3/256))= +1.37662e-002
data8 0x3f91a6b91ac73380, 0x3c7874daa716eb32 // log(1/frcpa(1+4/256))= +1.72376e-002
data8 0x3f95ba9a5d9ac000, 0x3cacbb84e08d78ac // log(1/frcpa(1+5/256))= +2.12196e-002
data8 0x3f99d2a807432580, 0x3cbcf80538b441e1 // log(1/frcpa(1+6/256))= +2.52177e-002
data8 0x3f9d6b2725979800, 0x3c6095e5c8f8f359 // log(1/frcpa(1+7/256))= +2.87291e-002
data8 0x3fa0c58fa19dfa80, 0x3cb4c5d4e9d0dda2 // log(1/frcpa(1+8/256))= +3.27573e-002
data8 0x3fa2954c78cbce00, 0x3caa932b860ab8d6 // log(1/frcpa(1+9/256))= +3.62953e-002
data8 0x3fa4a94d2da96c40, 0x3ca670452b76bbd5 // log(1/frcpa(1+10/256))= +4.03542e-002
data8 0x3fa67c94f2d4bb40, 0x3ca84104f9941798 // log(1/frcpa(1+11/256))= +4.39192e-002
data8 0x3fa85188b630f040, 0x3cb40a882cbf0153 // log(1/frcpa(1+12/256))= +4.74971e-002
data8 0x3faa6b8abe73af40, 0x3c988d46e25c9059 // log(1/frcpa(1+13/256))= +5.16017e-002
data8 0x3fac441e06f72a80, 0x3cae3e930a1a2a96 // log(1/frcpa(1+14/256))= +5.52072e-002
data8 0x3fae1e6713606d00, 0x3c8a796f6283b580 // log(1/frcpa(1+15/256))= +5.88257e-002
data8 0x3faffa6911ab9300, 0x3c5193070351e88a // log(1/frcpa(1+16/256))= +6.24574e-002
data8 0x3fb0ec139c5da600, 0x3c623f2a75eb992d // log(1/frcpa(1+17/256))= +6.61022e-002
data8 0x3fb1dbd2643d1900, 0x3ca649b2ef8927f0 // log(1/frcpa(1+18/256))= +6.97605e-002
data8 0x3fb2cc7284fe5f00, 0x3cbc5e86599513e2 // log(1/frcpa(1+19/256))= +7.34321e-002
data8 0x3fb3bdf5a7d1ee60, 0x3c90bd4bb69dada3 // log(1/frcpa(1+20/256))= +7.71173e-002
data8 0x3fb4b05d7aa012e0, 0x3c54e377c9b8a54f // log(1/frcpa(1+21/256))= +8.08161e-002
data8 0x3fb580db7ceb5700, 0x3c7fdb2f98354cde // log(1/frcpa(1+22/256))= +8.39975e-002
data8 0x3fb674f089365a60, 0x3cb9994c9d3301c1 // log(1/frcpa(1+23/256))= +8.77219e-002
data8 0x3fb769ef2c6b5680, 0x3caaec639db52a79 // log(1/frcpa(1+24/256))= +9.14602e-002
data8 0x3fb85fd927506a40, 0x3c9f9f99a3cf8e25 // log(1/frcpa(1+25/256))= +9.52125e-002
data8 0x3fb9335e5d594980, 0x3ca15c3abd47d99a // log(1/frcpa(1+26/256))= +9.84401e-002
data8 0x3fba2b0220c8e5e0, 0x3cb4ca639adf6fc3 // log(1/frcpa(1+27/256))= +1.02219e-001
data8 0x3fbb0004ac1a86a0, 0x3ca7cb81bf959a59 // log(1/frcpa(1+28/256))= +1.05469e-001
data8 0x3fbbf968769fca00, 0x3cb0c646c121418e // log(1/frcpa(1+29/256))= +1.09274e-001
data8 0x3fbccfedbfee13a0, 0x3ca0465fce24ab4b // log(1/frcpa(1+30/256))= +1.12548e-001
data8 0x3fbda727638446a0, 0x3c82803f4e2e6603 // log(1/frcpa(1+31/256))= +1.15832e-001
data8 0x3fbea3257fe10f60, 0x3cb986a3f2313d1a // log(1/frcpa(1+32/256))= +1.19677e-001
data8 0x3fbf7be9fedbfde0, 0x3c97d16a6a621cf4 // log(1/frcpa(1+33/256))= +1.22985e-001
data8 0x3fc02ab352ff25f0, 0x3c9cc6baad365600 // log(1/frcpa(1+34/256))= +1.26303e-001
data8 0x3fc097ce579d2040, 0x3cb9ba16d329440b // log(1/frcpa(1+35/256))= +1.29633e-001
data8 0x3fc1178e8227e470, 0x3cb7bc671683f8e6 // log(1/frcpa(1+36/256))= +1.33531e-001
data8 0x3fc185747dbecf30, 0x3c9d1116f66d2345 // log(1/frcpa(1+37/256))= +1.36885e-001
data8 0x3fc1f3b925f25d40, 0x3c8162c9ef939ac6 // log(1/frcpa(1+38/256))= +1.40250e-001
data8 0x3fc2625d1e6ddf50, 0x3caad3a1ec384fc3 // log(1/frcpa(1+39/256))= +1.43627e-001
data8 0x3fc2d1610c868130, 0x3cb3ad997036941b // log(1/frcpa(1+40/256))= +1.47015e-001
data8 0x3fc340c597411420, 0x3cbc2308262c7998 // log(1/frcpa(1+41/256))= +1.50414e-001
data8 0x3fc3b08b6757f2a0, 0x3cb2170d6cdf0526 // log(1/frcpa(1+42/256))= +1.53825e-001
data8 0x3fc40dfb08378000, 0x3c9bb453c4f7b685 // log(1/frcpa(1+43/256))= +1.56677e-001
data8 0x3fc47e74e8ca5f70, 0x3cb836a48fdfce9d // log(1/frcpa(1+44/256))= +1.60109e-001
data8 0x3fc4ef51f6466de0, 0x3ca07a43919aa64b // log(1/frcpa(1+45/256))= +1.63553e-001
data8 0x3fc56092e02ba510, 0x3ca85006899d97b0 // log(1/frcpa(1+46/256))= +1.67010e-001
data8 0x3fc5d23857cd74d0, 0x3ca30a5ba6e7abbe // log(1/frcpa(1+47/256))= +1.70478e-001
data8 0x3fc6313a37335d70, 0x3ca905586f0ac97e // log(1/frcpa(1+48/256))= +1.73377e-001
data8 0x3fc6a399dabbd380, 0x3c9b2c6657a96684 // log(1/frcpa(1+49/256))= +1.76868e-001
data8 0x3fc70337dd3ce410, 0x3cb50bc52f55cdd8 // log(1/frcpa(1+50/256))= +1.79786e-001
data8 0x3fc77654128f6120, 0x3cad2eb7c9a39efe // log(1/frcpa(1+51/256))= +1.83299e-001
data8 0x3fc7e9d82a0b0220, 0x3cba127e90393c01 // log(1/frcpa(1+52/256))= +1.86824e-001
data8 0x3fc84a6b759f5120, 0x3cbd7fd52079f706 // log(1/frcpa(1+53/256))= +1.89771e-001
data8 0x3fc8ab47d5f5a300, 0x3cbfae141751a3de // log(1/frcpa(1+54/256))= +1.92727e-001
data8 0x3fc91fe490965810, 0x3cb69cf30a1c319e // log(1/frcpa(1+55/256))= +1.96286e-001
data8 0x3fc981634011aa70, 0x3ca5bb3d208bc42a // log(1/frcpa(1+56/256))= +1.99261e-001
data8 0x3fc9f6c407089660, 0x3ca04d68658179a0 // log(1/frcpa(1+57/256))= +2.02843e-001
data8 0x3fca58e729348f40, 0x3c99f5411546c286 // log(1/frcpa(1+58/256))= +2.05838e-001
data8 0x3fcabb55c31693a0, 0x3cb9a5350eb327d5 // log(1/frcpa(1+59/256))= +2.08842e-001
data8 0x3fcb1e104919efd0, 0x3c18965fcce7c406 // log(1/frcpa(1+60/256))= +2.11855e-001
data8 0x3fcb94ee93e367c0, 0x3cb503716da45184 // log(1/frcpa(1+61/256))= +2.15483e-001
data8 0x3fcbf851c0675550, 0x3cbdf1b3f7ab5378 // log(1/frcpa(1+62/256))= +2.18516e-001
data8 0x3fcc5c0254bf23a0, 0x3ca7aab9ed0b1d7b // log(1/frcpa(1+63/256))= +2.21558e-001
data8 0x3fccc000c9db3c50, 0x3c92a7a2a850072a // log(1/frcpa(1+64/256))= +2.24609e-001
data8 0x3fcd244d99c85670, 0x3c9f6019120edf4c // log(1/frcpa(1+65/256))= +2.27670e-001
data8 0x3fcd88e93fb2f450, 0x3c6affb96815e081 // log(1/frcpa(1+66/256))= +2.30741e-001
data8 0x3fcdedd437eaef00, 0x3c72553595897976 // log(1/frcpa(1+67/256))= +2.33820e-001
data8 0x3fce530effe71010, 0x3c90913b020fa182 // log(1/frcpa(1+68/256))= +2.36910e-001
data8 0x3fceb89a1648b970, 0x3c837ba4045bfd25 // log(1/frcpa(1+69/256))= +2.40009e-001
data8 0x3fcf1e75fadf9bd0, 0x3cbcea6d13e0498d // log(1/frcpa(1+70/256))= +2.43117e-001
data8 0x3fcf84a32ead7c30, 0x3ca5e3a67b3c6d77 // log(1/frcpa(1+71/256))= +2.46235e-001
data8 0x3fcfeb2233ea07c0, 0x3cba0c6f0049c5a6 // log(1/frcpa(1+72/256))= +2.49363e-001
data8 0x3fd028f9c7035c18, 0x3cb0a30b06677ff6 // log(1/frcpa(1+73/256))= +2.52501e-001
data8 0x3fd05c8be0d96358, 0x3ca0f1c77ccb5865 // log(1/frcpa(1+74/256))= +2.55649e-001
data8 0x3fd085eb8f8ae790, 0x3cbd513f45fe7a97 // log(1/frcpa(1+75/256))= +2.58174e-001
data8 0x3fd0b9c8e32d1910, 0x3c927449047ca006 // log(1/frcpa(1+76/256))= +2.61339e-001
data8 0x3fd0edd060b78080, 0x3c89b52d8435f53e // log(1/frcpa(1+77/256))= +2.64515e-001
data8 0x3fd122024cf00638, 0x3cbdd976fabda4bd // log(1/frcpa(1+78/256))= +2.67701e-001
data8 0x3fd14be2927aecd0, 0x3cb02f90ad0bc471 // log(1/frcpa(1+79/256))= +2.70257e-001
data8 0x3fd180618ef18ad8, 0x3cbd003792c71a98 // log(1/frcpa(1+80/256))= +2.73461e-001
data8 0x3fd1b50bbe2fc638, 0x3ca9ae64c6403ead // log(1/frcpa(1+81/256))= +2.76675e-001
data8 0x3fd1df4cc7cf2428, 0x3cb43f0455f7e395 // log(1/frcpa(1+82/256))= +2.79254e-001
data8 0x3fd214456d0eb8d0, 0x3cb0fbd748d75d30 // log(1/frcpa(1+83/256))= +2.82487e-001
data8 0x3fd23ec5991eba48, 0x3c906edd746b77e2 // log(1/frcpa(1+84/256))= +2.85081e-001
data8 0x3fd2740d9f870af8, 0x3ca9802e6a00a670 // log(1/frcpa(1+85/256))= +2.88333e-001
data8 0x3fd29ecdabcdfa00, 0x3cacecef70890cfa // log(1/frcpa(1+86/256))= +2.90943e-001
data8 0x3fd2d46602adcce8, 0x3cb97911955f3521 // log(1/frcpa(1+87/256))= +2.94214e-001
data8 0x3fd2ff66b04ea9d0, 0x3cb12dabe191d1c9 // log(1/frcpa(1+88/256))= +2.96838e-001
data8 0x3fd335504b355a30, 0x3cbdf9139df924ec // log(1/frcpa(1+89/256))= +3.00129e-001
data8 0x3fd360925ec44f58, 0x3cb253e68977a1e3 // log(1/frcpa(1+90/256))= +3.02769e-001
data8 0x3fd38bf1c3337e70, 0x3cb3d283d2a2da21 // log(1/frcpa(1+91/256))= +3.05417e-001
data8 0x3fd3c25277333180, 0x3cadaa5b035eae27 // log(1/frcpa(1+92/256))= +3.08735e-001
data8 0x3fd3edf463c16838, 0x3cb983d680d3c108 // log(1/frcpa(1+93/256))= +3.11399e-001
data8 0x3fd419b423d5e8c0, 0x3cbc86dd921c139d // log(1/frcpa(1+94/256))= +3.14069e-001
data8 0x3fd44591e0539f48, 0x3c86a76d6dc2782e // log(1/frcpa(1+95/256))= +3.16746e-001
data8 0x3fd47c9175b6f0a8, 0x3cb59a2e013c6b5f // log(1/frcpa(1+96/256))= +3.20103e-001
data8 0x3fd4a8b341552b08, 0x3c93f1e86e468694 // log(1/frcpa(1+97/256))= +3.22797e-001
data8 0x3fd4d4f390890198, 0x3cbf5e4ea7c5105a // log(1/frcpa(1+98/256))= +3.25498e-001
data8 0x3fd501528da1f960, 0x3cbf58da53e9ad10 // log(1/frcpa(1+99/256))= +3.28206e-001
data8 0x3fd52dd06347d4f0, 0x3cb98a28cebf6eef // log(1/frcpa(1+100/256))= +3.30921e-001
data8 0x3fd55a6d3c7b8a88, 0x3c9c76b67c2d1fd4 // log(1/frcpa(1+101/256))= +3.33644e-001
data8 0x3fd5925d2b112a58, 0x3c9029616a4331b8 // log(1/frcpa(1+102/256))= +3.37058e-001
data8 0x3fd5bf406b543db0, 0x3c9fb8292ecfc820 // log(1/frcpa(1+103/256))= +3.39798e-001
data8 0x3fd5ec433d5c35a8, 0x3cb71a1229d17eec // log(1/frcpa(1+104/256))= +3.42545e-001
data8 0x3fd61965cdb02c18, 0x3cbba94fe1dbb8d2 // log(1/frcpa(1+105/256))= +3.45300e-001
data8 0x3fd646a84935b2a0, 0x3c9ee496d2c9ae57 // log(1/frcpa(1+106/256))= +3.48063e-001
data8 0x3fd6740add31de90, 0x3cb1da3a6c7a9dfd // log(1/frcpa(1+107/256))= +3.50833e-001
data8 0x3fd6a18db74a58c0, 0x3cb494c257add8dc // log(1/frcpa(1+108/256))= +3.53610e-001
data8 0x3fd6cf31058670e8, 0x3cb0b244a70a8da9 // log(1/frcpa(1+109/256))= +3.56396e-001
data8 0x3fd6f180e852f0b8, 0x3c9db7aefa866720 // log(1/frcpa(1+110/256))= +3.58490e-001
data8 0x3fd71f5d71b894e8, 0x3cbe91c4bf324957 // log(1/frcpa(1+111/256))= +3.61289e-001
data8 0x3fd74d5aefd66d58, 0x3cb06b3d9bfac023 // log(1/frcpa(1+112/256))= +3.64096e-001
data8 0x3fd77b79922bd378, 0x3cb727d8804491f4 // log(1/frcpa(1+113/256))= +3.66911e-001
data8 0x3fd7a9b9889f19e0, 0x3ca2ef22df5bc543 // log(1/frcpa(1+114/256))= +3.69734e-001
data8 0x3fd7d81b037eb6a0, 0x3cb8fd3ba07a7ece // log(1/frcpa(1+115/256))= +3.72565e-001
data8 0x3fd8069e33827230, 0x3c8bd1e25866e61a // log(1/frcpa(1+116/256))= +3.75404e-001
data8 0x3fd82996d3ef8bc8, 0x3ca5aab9f5928928 // log(1/frcpa(1+117/256))= +3.77538e-001
data8 0x3fd85855776dcbf8, 0x3ca56f33337789d6 // log(1/frcpa(1+118/256))= +3.80391e-001
data8 0x3fd8873658327cc8, 0x3cbb8ef0401db49d // log(1/frcpa(1+119/256))= +3.83253e-001
data8 0x3fd8aa75973ab8c8, 0x3cbb9961f509a680 // log(1/frcpa(1+120/256))= +3.85404e-001
data8 0x3fd8d992dc8824e0, 0x3cb220512a53732d // log(1/frcpa(1+121/256))= +3.88280e-001
data8 0x3fd908d2ea7d9510, 0x3c985f0e513bfb5c // log(1/frcpa(1+122/256))= +3.91164e-001
data8 0x3fd92c59e79c0e50, 0x3cb82e073fd30d63 // log(1/frcpa(1+123/256))= +3.93332e-001
data8 0x3fd95bd750ee3ed0, 0x3ca4aa7cdb6dd8a8 // log(1/frcpa(1+124/256))= +3.96231e-001
data8 0x3fd98b7811a3ee58, 0x3caa93a5b660893e // log(1/frcpa(1+125/256))= +3.99138e-001
data8 0x3fd9af47f33d4068, 0x3cac294b3b3190ba // log(1/frcpa(1+126/256))= +4.01323e-001
data8 0x3fd9df270c1914a0, 0x3cbe1a58fd0cd67e // log(1/frcpa(1+127/256))= +4.04245e-001
data8 0x3fda0325ed14fda0, 0x3cb1efa7950fb57e // log(1/frcpa(1+128/256))= +4.06442e-001
data8 0x3fda33440224fa78, 0x3c8915fe75e7d477 // log(1/frcpa(1+129/256))= +4.09379e-001
data8 0x3fda57725e80c380, 0x3ca72bd1062b1b7f // log(1/frcpa(1+130/256))= +4.11587e-001
data8 0x3fda87d0165dd198, 0x3c91f7845f58dbad // log(1/frcpa(1+131/256))= +4.14539e-001
data8 0x3fdaac2e6c03f890, 0x3cb6f237a911c509 // log(1/frcpa(1+132/256))= +4.16759e-001
data8 0x3fdadccc6fdf6a80, 0x3c90ddc4b7687169 // log(1/frcpa(1+133/256))= +4.19726e-001
data8 0x3fdb015b3eb1e790, 0x3c692dd7d90e1e8e // log(1/frcpa(1+134/256))= +4.21958e-001
data8 0x3fdb323a3a635948, 0x3c6f85655cbe14de // log(1/frcpa(1+135/256))= +4.24941e-001
data8 0x3fdb56fa04462908, 0x3c95252d841994de // log(1/frcpa(1+136/256))= +4.27184e-001
data8 0x3fdb881aa659bc90, 0x3caa53a745a3642f // log(1/frcpa(1+137/256))= +4.30182e-001
data8 0x3fdbad0bef3db160, 0x3cb32f2540dcc16a // log(1/frcpa(1+138/256))= +4.32437e-001
data8 0x3fdbd21297781c28, 0x3cbd8e891e106f1d // log(1/frcpa(1+139/256))= +4.34697e-001
data8 0x3fdc039236f08818, 0x3c809435af522ba7 // log(1/frcpa(1+140/256))= +4.37718e-001
data8 0x3fdc28cb1e4d32f8, 0x3cb3944752fbd81e // log(1/frcpa(1+141/256))= +4.39990e-001
data8 0x3fdc4e19b84723c0, 0x3c9a465260cd3fe5 // log(1/frcpa(1+142/256))= +4.42267e-001
data8 0x3fdc7ff9c74554c8, 0x3c92447d5b6ca369 // log(1/frcpa(1+143/256))= +4.45311e-001
data8 0x3fdca57b64e9db00, 0x3cb44344a8a00c82 // log(1/frcpa(1+144/256))= +4.47600e-001
data8 0x3fdccb130a5ceba8, 0x3cbefaddfb97b73f // log(1/frcpa(1+145/256))= +4.49895e-001
data8 0x3fdcf0c0d18f3268, 0x3cbd3e7bfee57898 // log(1/frcpa(1+146/256))= +4.52194e-001
data8 0x3fdd232075b5a200, 0x3c9222599987447c // log(1/frcpa(1+147/256))= +4.55269e-001
data8 0x3fdd490246defa68, 0x3cabafe9a767a80d // log(1/frcpa(1+148/256))= +4.57581e-001
data8 0x3fdd6efa918d25c8, 0x3cb58a2624e1c6fd // log(1/frcpa(1+149/256))= +4.59899e-001
data8 0x3fdd9509707ae528, 0x3cbdc3babce578e7 // log(1/frcpa(1+150/256))= +4.62221e-001
data8 0x3fddbb2efe92c550, 0x3cb0ac0943c434a4 // log(1/frcpa(1+151/256))= +4.64550e-001
data8 0x3fddee2f3445e4a8, 0x3cbba9d07ce820e8 // log(1/frcpa(1+152/256))= +4.67663e-001
data8 0x3fde148a1a2726c8, 0x3cb6537e3375b205 // log(1/frcpa(1+153/256))= +4.70004e-001
data8 0x3fde3afc0a49ff38, 0x3cbfed5518dbc20e // log(1/frcpa(1+154/256))= +4.72350e-001
data8 0x3fde6185206d5168, 0x3cb6572601f73d5c // log(1/frcpa(1+155/256))= +4.74702e-001
data8 0x3fde882578823d50, 0x3c9b24abd4584d1a // log(1/frcpa(1+156/256))= +4.77060e-001
data8 0x3fdeaedd2eac9908, 0x3cb0ceb5e4d2c8f7 // log(1/frcpa(1+157/256))= +4.79423e-001
data8 0x3fded5ac5f436be0, 0x3ca72f21f1f5238e // log(1/frcpa(1+158/256))= +4.81792e-001
data8 0x3fdefc9326d16ab8, 0x3c85081a1639a45c // log(1/frcpa(1+159/256))= +4.84166e-001
data8 0x3fdf2391a21575f8, 0x3cbf11015bdd297a // log(1/frcpa(1+160/256))= +4.86546e-001
data8 0x3fdf4aa7ee031928, 0x3cb3795bc052a2d1 // log(1/frcpa(1+161/256))= +4.88932e-001
data8 0x3fdf71d627c30bb0, 0x3c35c61f0f5a88f3 // log(1/frcpa(1+162/256))= +4.91323e-001
data8 0x3fdf991c6cb3b378, 0x3c97d99419be6028 // log(1/frcpa(1+163/256))= +4.93720e-001
data8 0x3fdfc07ada69a908, 0x3cbfe9341ded70b1 // log(1/frcpa(1+164/256))= +4.96123e-001
data8 0x3fdfe7f18eb03d38, 0x3cb85718a640c33f // log(1/frcpa(1+165/256))= +4.98532e-001
data8 0x3fe007c053c5002c, 0x3cb3addc9c065f09 // log(1/frcpa(1+166/256))= +5.00946e-001
data8 0x3fe01b942198a5a0, 0x3c9d5aa4c77da6ac // log(1/frcpa(1+167/256))= +5.03367e-001
data8 0x3fe02f74400c64e8, 0x3cb5a0ee4450ef52 // log(1/frcpa(1+168/256))= +5.05793e-001
data8 0x3fe04360be7603ac, 0x3c9dd00c35630fe0 // log(1/frcpa(1+169/256))= +5.08225e-001
data8 0x3fe05759ac47fe30, 0x3cbd063e1f0bd82c // log(1/frcpa(1+170/256))= +5.10663e-001
data8 0x3fe06b5f1911cf50, 0x3cae8da674af5289 // log(1/frcpa(1+171/256))= +5.13107e-001
data8 0x3fe078bf0533c568, 0x3c62241edf5fd1f7 // log(1/frcpa(1+172/256))= +5.14740e-001
data8 0x3fe08cd9687e7b0c, 0x3cb3007febcca227 // log(1/frcpa(1+173/256))= +5.17194e-001
data8 0x3fe0a10074cf9018, 0x3ca496e84603816b // log(1/frcpa(1+174/256))= +5.19654e-001
data8 0x3fe0b5343a234474, 0x3cb46098d14fc90a // log(1/frcpa(1+175/256))= +5.22120e-001
data8 0x3fe0c974c89431cc, 0x3cac0a7cdcbb86c6 // log(1/frcpa(1+176/256))= +5.24592e-001
data8 0x3fe0ddc2305b9884, 0x3cb2f753210410ff // log(1/frcpa(1+177/256))= +5.27070e-001
data8 0x3fe0eb524bafc918, 0x3c88affd6682229e // log(1/frcpa(1+178/256))= +5.28726e-001
data8 0x3fe0ffb54213a474, 0x3cadeefbab9af993 // log(1/frcpa(1+179/256))= +5.31214e-001
data8 0x3fe114253da97d9c, 0x3cbaf1c2b8bc160a // log(1/frcpa(1+180/256))= +5.33709e-001
data8 0x3fe128a24f1d9afc, 0x3cb9cf4df375e650 // log(1/frcpa(1+181/256))= +5.36210e-001
data8 0x3fe1365252bf0864, 0x3c985a621d4be111 // log(1/frcpa(1+182/256))= +5.37881e-001
data8 0x3fe14ae558b4a92c, 0x3ca104c4aa8977d1 // log(1/frcpa(1+183/256))= +5.40393e-001
data8 0x3fe15f85a19c7658, 0x3cbadf26e540f375 // log(1/frcpa(1+184/256))= +5.42910e-001
data8 0x3fe16d4d38c119f8, 0x3cb3aea11caec416 // log(1/frcpa(1+185/256))= +5.44592e-001
data8 0x3fe18203c20dd130, 0x3cba82d1211d1d6d // log(1/frcpa(1+186/256))= +5.47121e-001
data8 0x3fe196c7bc4b1f38, 0x3cb6267acc4f4f4a // log(1/frcpa(1+187/256))= +5.49656e-001
data8 0x3fe1a4a738b7a33c, 0x3c858930213c987d // log(1/frcpa(1+188/256))= +5.51349e-001
data8 0x3fe1b981c0c9653c, 0x3c9bc2a4a30f697b // log(1/frcpa(1+189/256))= +5.53895e-001
data8 0x3fe1ce69e8bb1068, 0x3cb7ae6199cf2a00 // log(1/frcpa(1+190/256))= +5.56447e-001
data8 0x3fe1dc619de06944, 0x3c6b50bb38388177 // log(1/frcpa(1+191/256))= +5.58152e-001
data8 0x3fe1f160a2ad0da0, 0x3cbd05b2778a5e1d // log(1/frcpa(1+192/256))= +5.60715e-001
data8 0x3fe2066d7740737c, 0x3cb32e828f9c6bd6 // log(1/frcpa(1+193/256))= +5.63285e-001
data8 0x3fe2147dba47a390, 0x3cbd579851b8b672 // log(1/frcpa(1+194/256))= +5.65001e-001
data8 0x3fe229a1bc5ebac0, 0x3cbb321be5237ce8 // log(1/frcpa(1+195/256))= +5.67582e-001
data8 0x3fe237c1841a502c, 0x3cb3b56e0915ea64 // log(1/frcpa(1+196/256))= +5.69306e-001
data8 0x3fe24cfce6f80d98, 0x3cb34a4d1a422919 // log(1/frcpa(1+197/256))= +5.71898e-001
data8 0x3fe25b2c55cd5760, 0x3cb237401ea5015e // log(1/frcpa(1+198/256))= +5.73630e-001
data8 0x3fe2707f4d5f7c40, 0x3c9d30f20acc8341 // log(1/frcpa(1+199/256))= +5.76233e-001
data8 0x3fe285e0842ca380, 0x3cbc4d866d5f21c0 // log(1/frcpa(1+200/256))= +5.78842e-001
data8 0x3fe294294708b770, 0x3cb85e14d5dc54fa // log(1/frcpa(1+201/256))= +5.80586e-001
data8 0x3fe2a9a2670aff0c, 0x3c7e6f8f468bbf91 // log(1/frcpa(1+202/256))= +5.83207e-001
data8 0x3fe2b7fb2c8d1cc0, 0x3c930ffcf63c8b65 // log(1/frcpa(1+203/256))= +5.84959e-001
data8 0x3fe2c65a6395f5f4, 0x3ca0afe20b53d2d2 // log(1/frcpa(1+204/256))= +5.86713e-001
data8 0x3fe2dbf557b0df40, 0x3cb646be1188fbc9 // log(1/frcpa(1+205/256))= +5.89350e-001
data8 0x3fe2ea64c3f97654, 0x3c96516fa8df33b2 // log(1/frcpa(1+206/256))= +5.91113e-001
data8 0x3fe3001823684d70, 0x3cb96d64e16d1360 // log(1/frcpa(1+207/256))= +5.93762e-001
data8 0x3fe30e97e9a8b5cc, 0x3c98ef96bc97cca0 // log(1/frcpa(1+208/256))= +5.95531e-001
data8 0x3fe32463ebdd34e8, 0x3caef1dc9a56c1bf // log(1/frcpa(1+209/256))= +5.98192e-001
data8 0x3fe332f4314ad794, 0x3caa4f0ac5d5fa11 // log(1/frcpa(1+210/256))= +5.99970e-001
data8 0x3fe348d90e7464cc, 0x3cbe7889f0516acd // log(1/frcpa(1+211/256))= +6.02643e-001
data8 0x3fe35779f8c43d6c, 0x3ca96bbab7245411 // log(1/frcpa(1+212/256))= +6.04428e-001
data8 0x3fe36621961a6a98, 0x3ca31f32262db9fb // log(1/frcpa(1+213/256))= +6.06217e-001
data8 0x3fe37c299f3c3668, 0x3cb15c72c107ee29 // log(1/frcpa(1+214/256))= +6.08907e-001
data8 0x3fe38ae2171976e4, 0x3cba42a2554b2dd4 // log(1/frcpa(1+215/256))= +6.10704e-001
data8 0x3fe399a157a603e4, 0x3cb99c62286d8919 // log(1/frcpa(1+216/256))= +6.12504e-001
data8 0x3fe3afccfe77b9d0, 0x3ca11048f96a43bd // log(1/frcpa(1+217/256))= +6.15210e-001
data8 0x3fe3be9d503533b4, 0x3ca4022f47588c3e // log(1/frcpa(1+218/256))= +6.17018e-001
data8 0x3fe3cd7480b4a8a0, 0x3cb4ba7afc2dc56a // log(1/frcpa(1+219/256))= +6.18830e-001
data8 0x3fe3e3c43918f76c, 0x3c859673d064b8ba // log(1/frcpa(1+220/256))= +6.21554e-001
data8 0x3fe3f2acb27ed6c4, 0x3cb55c6b452a16a8 // log(1/frcpa(1+221/256))= +6.23373e-001
data8 0x3fe4019c2125ca90, 0x3cb8c367879c5a31 // log(1/frcpa(1+222/256))= +6.25197e-001
data8 0x3fe4181061389720, 0x3cb2c17a79c5cc6c // log(1/frcpa(1+223/256))= +6.27937e-001
data8 0x3fe42711518df544, 0x3ca5f38d47012fc5 // log(1/frcpa(1+224/256))= +6.29769e-001
data8 0x3fe436194e12b6bc, 0x3cb9854d65a9b426 // log(1/frcpa(1+225/256))= +6.31604e-001
data8 0x3fe445285d68ea68, 0x3ca3ff9b3a81cd81 // log(1/frcpa(1+226/256))= +6.33442e-001
data8 0x3fe45bcc464c8938, 0x3cb0a2d8011a6c05 // log(1/frcpa(1+227/256))= +6.36206e-001
data8 0x3fe46aed21f117fc, 0x3c8a2be41f8e9f3d // log(1/frcpa(1+228/256))= +6.38053e-001
data8 0x3fe47a1527e8a2d0, 0x3cba4a83594fab09 // log(1/frcpa(1+229/256))= +6.39903e-001
data8 0x3fe489445efffcc8, 0x3cbf306a23dcbcde // log(1/frcpa(1+230/256))= +6.41756e-001
data8 0x3fe4a018bcb69834, 0x3ca46c9285029fd1 // log(1/frcpa(1+231/256))= +6.44543e-001
data8 0x3fe4af5a0c9d65d4, 0x3cbbc1db897580e3 // log(1/frcpa(1+232/256))= +6.46405e-001
data8 0x3fe4bea2a5bdbe84, 0x3cb84d880d7ef775 // log(1/frcpa(1+233/256))= +6.48271e-001
data8 0x3fe4cdf28f10ac44, 0x3cb3ec4b7893ce1f // log(1/frcpa(1+234/256))= +6.50140e-001
data8 0x3fe4dd49cf994058, 0x3c897224d59d3408 // log(1/frcpa(1+235/256))= +6.52013e-001
data8 0x3fe4eca86e64a680, 0x3cbccf620f24f0cd // log(1/frcpa(1+236/256))= +6.53889e-001
data8 0x3fe503c43cd8eb68, 0x3c3f872c65971084 // log(1/frcpa(1+237/256))= +6.56710e-001
data8 0x3fe513356667fc54, 0x3cb9ca64cc3d52c8 // log(1/frcpa(1+238/256))= +6.58595e-001
data8 0x3fe522ae0738a3d4, 0x3cbe708164c75968 // log(1/frcpa(1+239/256))= +6.60483e-001
data8 0x3fe5322e26867854, 0x3cb9988ba4aea615 // log(1/frcpa(1+240/256))= +6.62376e-001
data8 0x3fe541b5cb979808, 0x3ca1662e3a6b95f5 // log(1/frcpa(1+241/256))= +6.64271e-001
data8 0x3fe55144fdbcbd60, 0x3cb3acd4ca45c1e0 // log(1/frcpa(1+242/256))= +6.66171e-001
data8 0x3fe560dbc45153c4, 0x3cb4988947959fed // log(1/frcpa(1+243/256))= +6.68074e-001
data8 0x3fe5707a26bb8c64, 0x3cb3017fe6607ba9 // log(1/frcpa(1+244/256))= +6.69980e-001
data8 0x3fe587f60ed5b8fc, 0x3cbe7a3266366ed4 // log(1/frcpa(1+245/256))= +6.72847e-001
data8 0x3fe597a7977c8f30, 0x3ca1e12b9959a90e // log(1/frcpa(1+246/256))= +6.74763e-001
data8 0x3fe5a760d634bb88, 0x3cb7c365e53d9602 // log(1/frcpa(1+247/256))= +6.76682e-001
data8 0x3fe5b721d295f10c, 0x3cb716c2551ccbf0 // log(1/frcpa(1+248/256))= +6.78605e-001
data8 0x3fe5c6ea94431ef8, 0x3ca02b2ed0e28261 // log(1/frcpa(1+249/256))= +6.80532e-001
data8 0x3fe5d6bb22ea86f4, 0x3caf43a8bbb2f974 // log(1/frcpa(1+250/256))= +6.82462e-001
data8 0x3fe5e6938645d38c, 0x3cbcedc98821b333 // log(1/frcpa(1+251/256))= +6.84397e-001
data8 0x3fe5f673c61a2ed0, 0x3caa385eef5f2789 // log(1/frcpa(1+252/256))= +6.86335e-001
data8 0x3fe6065bea385924, 0x3cb11624f165c5b4 // log(1/frcpa(1+253/256))= +6.88276e-001
data8 0x3fe6164bfa7cc068, 0x3cbad884f87073fa // log(1/frcpa(1+254/256))= +6.90222e-001
data8 0x3fe62643fecf9740, 0x3cb78c51da12f4df // 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(pow)
// 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
nop.i 999
}
{ .mfi
ld8 pow_AD_P = [pow_AD_P]
fma.s1 POW_NORM_X = f8,f1,f0
mov pow_GR_exp_2tom8 = 0xFFF7
}
;;
// p13 = TRUE ==> X is unorm
// DOUBLE 0x10033 exponent limit at which y is an integer
{ .mfi
nop.m 999
fclass.m p13,p0 = f8, 0x0b // Test for x unorm
addl pow_GR_10033 = 0x10033, r0
}
{ .mfi
mov pow_GR_16ones = 0xFFFF
fma.s1 POW_NORM_Y = f9,f1,f0
nop.i 999
}
;;
// p14 = TRUE ==> X is ZERO
{ .mfi
adds pow_AD_Tt = pow_Tt - pow_table_P, pow_AD_P
fclass.m p14,p0 = f8, 0x07
and pow_GR_exp_X = pow_GR_signexp_X, pow_GR_17ones
}
{ .mfi
adds pow_AD_Q = pow_table_Q - pow_table_P, pow_AD_P
nop.f 999
nop.i 999
}
;;
{ .mfi
ldfe POW_P5 = [pow_AD_P], 16
fcmp.lt.s1 p8,p9 = f8, f0 // Test for x<0
nop.i 999
}
{ .mib
ldfe POW_P4 = [pow_AD_Q], 16
sub pow_GR_true_exp_X = pow_GR_exp_X, pow_GR_16ones
(p13) br.cond.spnt POW_X_DENORM
}
;;
// Continue normal and denormal paths here
POW_COMMON:
// p11 = TRUE ==> Y is a NAN
{ .mfi
ldfe POW_P3 = [pow_AD_P], 16
fclass.m p11,p0 = f9, 0xc3
nop.i 999
}
{ .mfi
ldfe POW_P2 = [pow_AD_Q], 16
nop.f 999
mov pow_GR_y_zero = 0
}
;;
// Note POW_Xm1 and POW_r1 are used interchangably
{ .mfi
alloc r32=ar.pfs,2,19,4,0
fms.s1 POW_r = POW_B, POW_NORM_X,f1
nop.i 999
}
{ .mfi
setf.sig POW_int_K = pow_GR_true_exp_X
(p8) fnma.s1 POW_Xm1 = POW_Xp1,f1,f0
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
shl pow_GR_offset = pow_GR_sig_X, 1
}
{ .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
}
;;
// Get exponent of |x|-1 to use in comparison to 2^-8
{ .mfi
getf.exp pow_GR_signexp_Xm1 = POW_Xm1
fcvt.fx.s1 POW_int_Y = POW_NORM_Y
shr.u pow_GR_offset = pow_GR_offset,56
}
;;
// p11 = TRUE ==> X is a NAN
{ .mfi
ldfpd POW_log2_hi, POW_log2_lo = [pow_AD_Q], 16
fclass.m p11,p0 = f8, 0xc3
shladd pow_AD_Tt = pow_GR_offset, 4, pow_AD_Tt
}
{ .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
ldfpd POW_Q2, POW_Q3 = [pow_AD_P], 16
fma.s1 POW_G = f0,f0,f0 // G=0 in case |x| near 1
and pow_GR_exp_Xm1 = pow_GR_signexp_Xm1, pow_GR_17ones
}
;;
// Determine if we will use the |x| near 1 path (p6) or normal path (p7)
{ .mfi
getf.exp pow_GR_signexp_Y = POW_NORM_Y
nop.f 999
cmp.lt p6,p7 = pow_GR_exp_Xm1, pow_GR_exp_2tom8
}
{ .mfb
ldfpd POW_T, POW_Tt = [pow_AD_Tt], 16
fma.s1 POW_rsq = POW_r, POW_r,f0
(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 and r1*r1 to rsq
{ .mfi
ldfpd POW_Q0_half, POW_Q1 = [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
ldfpd POW_Q4, POW_RSHF = [pow_AD_P], 16
(p7) fma.s1 POW_v6 = POW_r, POW_P5, POW_P4
nop.i 999
}
{ .mfi
mov pow_GR_exp_2toM63 = 0xffc0 // Exponent of 2^-63
(p6) fma.s1 POW_v6 = POW_r1, POW_P5, POW_P4
nop.i 999
}
;;
{ .mfi
setf.exp POW_2toM63 = pow_GR_exp_2toM63 // Form 2^-63 for test of q
(p7) fma.s1 POW_v4 = POW_P3, POW_r, POW_P2
nop.i 999
}
{ .mfi
nop.m 999
(p6) fma.s1 POW_v4 = POW_P3, POW_r1, POW_P2
nop.i 999
}
;;
{ .mfi
nop.m 999
fcvt.xf POW_K = POW_int_K
nop.i 999
}
;;
{ .mfi
getf.sig pow_GR_sig_int_Y = POW_int_Y
fnma.s1 POW_twoV = POW_NORM_Y, POW_rsq,f0
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
fma.s1 POW_U = POW_NORM_Y,POW_r,f0
(p12) br.cond.spnt POW_Y_0 // Branch if y=zero, x not zero or nan
}
;;
// p11 = TRUE ==> X is NEGATIVE but not inf
{ .mfi
ldfe POW_log2_by_128_lo = [pow_AD_P], 16
fclass.m p11,p0 = f8, 0x1a
nop.i 999
}
{ .mfi
ldfe POW_log2_by_128_hi = [pow_AD_Q], 16
fma.s1 POW_v2 = POW_P1, POW_r, POW_P0
nop.i 999
}
;;
{ .mfi
nop.m 999
fcvt.xf POW_float_int_Y = POW_int_Y
nop.i 999
}
{ .mfi
nop.m 999
fma.s1 POW_v3 = POW_v6, POW_rsq, POW_v4
adds pow_AD_tbl1 = pow_tbl1 - pow_Tt, pow_AD_Q
}
;;
{ .mfi
nop.m 999
(p7) fma.s1 POW_delta = POW_K, POW_log2_lo, POW_Tt
nop.i 999
}
{ .mfi
nop.m 999
(p7) fma.s1 POW_G = POW_K, POW_log2_hi, POW_T
adds pow_AD_tbl2 = pow_tbl2 - pow_tbl1, pow_AD_tbl1
}
;;
{ .mfi
nop.m 999
fms.s1 POW_e2 = POW_NORM_Y, POW_r, POW_U
nop.i 999
}
{ .mfi
nop.m 999
fma.s1 POW_Z2 = POW_twoV, POW_Q0_half, POW_U
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_v3, POW_v2
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
fma.s1 POW_Z1 = POW_NORM_Y, POW_G, f0
(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
}
;;
// By adding RSHF (1.1000...*2^63) we put integer part in rightmost significand
{ .mfi
nop.m 999
fma.s1 POW_W2 = POW_Z2, POW_inv_log2_by_128, POW_RSHF
nop.i 999
}
{ .mfi
nop.m 999
fms.s1 POW_UmZ2 = POW_U, f1, POW_Z2
nop.i 999
}
;;
{ .mfi
nop.m 999
fma.s1 POW_e3 = POW_NORM_Y, POW_delta, f0
nop.i 999
}
;;
{ .mfi
nop.m 999
fma.s1 POW_Z3 = POW_p, POW_Yrcub, f0
nop.i 999
}
{ .mfi
nop.m 999
fma.s1 POW_GY_Z2 = POW_G, POW_NORM_Y, POW_Z2
nop.i 999
}
;;
// By adding RSHF (1.1000...*2^63) we put integer part in rightmost significand
{ .mfi
nop.m 999
fms.s1 POW_e1 = POW_NORM_Y, POW_G, POW_Z1
nop.i 999
}
{ .mfi
nop.m 999
fma.s1 POW_W1 = POW_Z1, 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
}
;;
// By subtracting RSHF we get rounded integer POW_N2float
{ .mfi
nop.m 999
fms.s1 POW_N2float = POW_W2, f1, POW_RSHF
nop.i 999
}
{ .mfi
nop.m 999
fma.s1 POW_UmZ2pV = POW_twoV,POW_Q0_half,POW_UmZ2
nop.i 999
}
;;
{ .mfi
nop.m 999
fma.s1 POW_Z3sq = POW_Z3, POW_Z3, f0
nop.i 999
}
{ .mfi
nop.m 999
fma.s1 POW_v4 = POW_Z3, POW_Q3, POW_Q2
nop.i 999
}
;;
// Extract rounded integer from rightmost significand of POW_W2
// By subtracting RSHF we get rounded integer POW_N1float
{ .mfi
getf.sig pow_GR_int_W2 = POW_W2
fms.s1 POW_N1float = POW_W1, f1, POW_RSHF
nop.i 999
}
{ .mfi
nop.m 999
fma.s1 POW_v2 = POW_Z3, POW_Q1, POW_Q0_half
nop.i 999
}
;;
{ .mfi
nop.m 999
fnma.s1 POW_s2 = POW_N2float, POW_log2_by_128_hi, POW_Z2
nop.i 999
}
{ .mfi
nop.m 999
fma.s1 POW_e2 = POW_e2,f1,POW_UmZ2pV
nop.i 999
}
;;
// Extract rounded integer from rightmost significand of POW_W1
// Test if x inf
{ .mfi
getf.sig pow_GR_int_W1 = POW_W1
fclass.m p15,p0 = POW_NORM_X, 0x23
nop.i 999
}
{ .mfb
nop.m 999
fnma.s1 POW_f2 = POW_N2float, POW_log2_by_128_lo, f1
(p12) br.cond.spnt POW_X_NEG_Y_NONINT // Branch if x neg, y not integer
}
;;
// p11 = TRUE ==> X 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 p11,p0 = POW_NORM_X, f1
(p10) tbit.nz.unc p12,p0 = pow_GR_sig_int_Y,0
}
{ .mfi
nop.m 999
fma.s1 POW_v3 = POW_Z3sq, POW_Q4, POW_v4
nop.i 999
}
;;
{ .mfi
nop.m 999
fnma.s1 POW_f1 = POW_N1float, POW_log2_by_128_lo, f1
nop.i 999
}
{ .mfb
nop.m 999
fnma.s1 POW_s1 = POW_N1float, POW_log2_by_128_hi, POW_Z1
(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
}
{ .mfi
nop.m 999
fma.s1 POW_pYrcub_e3 = POW_p, POW_Yrcub, POW_e3
nop.i 999
}
;;
{ .mfi
nop.m 999
fcmp.eq.s1 p7,p0 = POW_NORM_Y, f1 // Test for y=1.0
nop.i 999
}
{ .mfi
nop.m 999
fma.s1 POW_e12 = POW_e1,f1,POW_e2
nop.i 999
}
;;
{ .mfi
add pow_GR_int_N = pow_GR_int_W1, pow_GR_int_W2
(p11) fma.d.s0 f8 = f1,f1,f0 // If x=1, result is +1
nop.i 999
}
{ .mib
(p12) mov pow_GR_xneg_yodd = 1
nop.i 999
(p11) br.ret.spnt b0 // Early exit if x=1.0, result is +1
}
;;
{ .mfi
and pow_GR_index1 = 0x0f, pow_GR_int_N
fma.s1 POW_q = POW_Z3sq, POW_v3, POW_v2
shr pow_int_GR_M = pow_GR_int_N, 7 // M = N/128
}
{ .mib
and pow_GR_index2 = 0x70, pow_GR_int_N
cmp.eq p6, p0 = pow_GR_xneg_yodd, r0
(p7) br.ret.spnt b0 // Early exit if y=1.0, result is x
}
;;
{ .mfi
shladd pow_AD_T1 = pow_GR_index1, 4, pow_AD_tbl1
fma.s1 POW_s = POW_s1, f1, POW_s2
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_f12 = POW_f1, POW_f2,f0
and pow_GR_exp_Y_Gpr = pow_GR_signexp_Y_Gpr, pow_GR_17ones
}
;;
{ .mmi
ldfe POW_T1 = [pow_AD_T1]
ldfe POW_T2 = [pow_AD_T2]
sub pow_GR_true_exp_Y_Gpr = pow_GR_exp_Y_Gpr, pow_GR_16ones
}
;;
{ .mfi
setf.exp POW_2M = pow_int_GR_M
fma.s1 POW_e123 = POW_e12, f1, POW_e3
nop.i 999
}
{ .mfb
(p6) cmp.gt p6, p0 = -11, pow_GR_true_exp_Y_Gpr
fma.s1 POW_d = POW_GY_Z2, f1, POW_pYrcub_e3
(p6) br.cond.spnt POW_NEAR_ONE // branch if |y*log(x)| < 2^(-11)
}
;;
{ .mfi
nop.m 999
fma.s1 POW_q = POW_Z3sq, POW_q, POW_Z3
nop.i 999
}
;;
// p8 TRUE ==> |Y(G + r)| >= 10
// double
// -2^10 -2^9 2^9 2^10
// -----+-----+----+ ... +-----+-----+-----
// p8 | p9 | p8
// | | p10 | |
// Form signexp of constants to indicate overflow
{ .mfi
mov pow_GR_big_pos = 0x103ff
fma.s1 POW_ssq = POW_s, POW_s, f0
cmp.le p8,p9 = 10, pow_GR_true_exp_Y_Gpr
}
{ .mfi
mov pow_GR_big_neg = 0x303ff
fma.s1 POW_v4 = POW_s, POW_Q3, POW_Q2
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
{ .mfi
setf.exp POW_big_pos = pow_GR_big_pos
fma.s1 POW_v2 = POW_s, POW_Q1, POW_Q0_half
(p9) cmp.le.unc p0,p10 = 9, pow_GR_true_exp_Y_Gpr
}
{ .mfb
setf.exp POW_big_neg = pow_GR_big_neg
fma.s1 POW_1ps = f1,f1,POW_s
(p8) br.cond.spnt POW_OVER_UNDER_X_NOT_INF
}
;;
// f123 = f12*(e123+1) = f12*e123+f12
{ .mfi
nop.m 999
fma.s1 POW_f123 = POW_e123,POW_f12,POW_f12
nop.i 999
}
;;
{ .mfi
nop.m 999
fma.s1 POW_T1T2 = POW_T1, POW_T2, f0
nop.i 999
}
{ .mfi
nop.m 999
fma.s1 POW_v3 = POW_ssq, POW_Q4, POW_v4
cmp.ne p12,p13 = pow_GR_xneg_yodd, r0
}
;;
{ .mfi
nop.m 999
fma.s1 POW_v21ps = POW_ssq, POW_v2, POW_1ps
nop.i 999
}
{ .mfi
nop.m 999
fma.s1 POW_s4 = POW_ssq, POW_ssq, f0
nop.i 999
}
;;
{ .mfi
nop.m 999
(p12) fnma.s1 POW_A = POW_2M, POW_f123, f0
nop.i 999
}
{ .mfi
nop.m 999
(p13) fma.s1 POW_A = POW_2M, POW_f123, f0
cmp.eq p14,p11 = r0,r0 // Initialize p14 on, p11 off
}
;;
{ .mfi
nop.m 999
fmerge.s POW_abs_q = f0, POW_q // Form |q| so can test its size
nop.i 999
}
;;
{ .mfi
(p10) cmp.eq p0,p14 = r0,r0 // Turn off p14 if no overflow
fma.s1 POW_es = POW_s4, POW_v3, POW_v21ps
nop.i 999
}
{ .mfi
nop.m 999
fma.s1 POW_A = POW_A, POW_T1T2, f0
nop.i 999
}
;;
{ .mfi
// Test for |q| < 2^-63. If so then reverse last two steps of the result
// to avoid monotonicity problems for results near 1.0 in round up/down/zero.
// p11 will be set if need to reverse the order, p14 if not.
nop.m 999
(p10) fcmp.lt.s0 p11,p14 = POW_abs_q, POW_2toM63 // Test |q| <2^-63
nop.i 999
}
;;
.pred.rel "mutex",p11,p14
{ .mfi
nop.m 999
(p14) fma.s1 POW_A = POW_A, POW_es, f0
nop.i 999
}
{ .mfi
nop.m 999
(p11) fma.s1 POW_A = POW_A, POW_q, POW_A
nop.i 999
}
;;
// Dummy op to set inexact if |q| < 2^-63
{ .mfi
nop.m 999
(p11) fma.d.s0 POW_tmp = POW_A, POW_q, POW_A
nop.i 999
}
;;
{ .mfi
nop.m 999
(p14) fma.d.s0 f8 = POW_A, POW_q, POW_A
nop.i 999
}
{ .mfb
nop.m 999
(p11) fma.d.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.d.s2 POW_wre_urm_f8 = POW_A, POW_q, POW_A
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 = 24
(p7) br.cond.spnt __libm_error_region // Branch if overflow
br.ret.sptk b0 // Exit if did not overflow
}
;;
// Here if |y*log(x)| < 2^(-11)
// pow(x,y) ~ exp(d) ~ 1 + d + 0.5*d^2 + Q1*d^3 + Q2*d^4, where d = y*log(x)
.align 32
POW_NEAR_ONE:
{ .mfi
nop.m 999
fma.s1 POW_d2 = POW_d, POW_d, f0
nop.i 999
}
;;
{ .mfi
nop.m 999
fma.s1 POW_poly_d_hi = POW_d, POW_Q0_half, f1
nop.i 999
}
{ .mfi
nop.m 999
fma.s1 POW_poly_d_lo = POW_d, POW_Q2, POW_Q1
nop.i 999
}
;;
{ .mfi
nop.m 999
fma.s1 POW_poly_d = POW_d2, POW_poly_d_lo, POW_poly_d_hi
nop.i 999
}
;;
{ .mfb
nop.m 999
fma.d.s0 f8 = POW_d, POW_poly_d, f1
br.ret.sptk b0 // exit function for arguments |y*log(x)| < 2^(-11)
}
;;
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
// Put in s2 (td set, ftz set)
{ .mfi
nop.m 999
fsetc.s2 0x7F,0x41
nop.i 999
}
;;
{ .mfi
nop.m 999
fma.d.s2 POW_ftz_urm_f8 = POW_A, POW_q, POW_A
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
}
;;
{ .mbb
(p7) mov pow_GR_tag = 25
(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
}
;;
{ .mmi
getf.sig pow_GR_sig_X = POW_NORM_X
;;
and pow_GR_exp_X = pow_GR_signexp_X, pow_GR_17ones
nop.i 999
}
;;
{ .mib
sub pow_GR_true_exp_X = pow_GR_exp_X, pow_GR_16ones
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.d.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.d.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.d.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 = 26
fma.d.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 = 27
(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.d.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.d.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 = 28
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.d.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 = 29
fma.d.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.d.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.d.s0 f8 = f1,f1,f0
nop.i 999
}
{ .mfb
nop.m 999
(p9) fma.d.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 = 24
(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 = 25
(p9) setf.exp POW_tmp = pow_GR_one
nop.i 999
}
;;
{ .mfi
nop.m 999
fma.d.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.d.s0 f8 = POW_tmp, POW_tmp, f0
nop.i 999
}
;;
GLOBAL_LIBM_END(pow)
libm_alias_double_other (pow, pow)
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
stfd [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
stfd [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
stfd [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
ldfd 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#