| |
| /* |
| * Copyright (C) 2008-2009 Advanced Micro Devices, Inc. All Rights Reserved. |
| * |
| * This file is part of libacml_mv. |
| * |
| * libacml_mv is free software; you can redistribute it and/or |
| * modify it under the terms of the GNU Lesser General Public |
| * License as published by the Free Software Foundation; either |
| * version 2.1 of the License, or (at your option) any later version. |
| * |
| * libacml_mv is distributed in the hope that it will be useful, |
| * but WITHOUT ANY WARRANTY; without even the implied warranty of |
| * MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU |
| * Lesser General Public License for more details. |
| * |
| * You should have received a copy of the GNU Lesser General Public |
| * License along with libacml_mv. If not, see |
| * <http://www.gnu.org/licenses/>. |
| * |
| */ |
| |
| |
| #include "../inc/libm_amd.h" |
| #include "../inc/libm_util_amd.h" |
| |
| #define USE_NAN_WITH_FLAGS |
| #define USE_VAL_WITH_FLAGS |
| #define USE_INFINITY_WITH_FLAGS |
| #define USE_HANDLE_ERROR |
| #include "../inc/libm_inlines_amd.h" |
| #undef USE_NAN_WITH_FLAGS |
| #undef USE_VAL_WITH_FLAGS |
| #undef USE_INFINITY_WITH_FLAGS |
| #undef USE_HANDLE_ERROR |
| |
| #include "../inc/libm_errno_amd.h" |
| |
| #ifndef WINDOWS |
| /* Deal with errno for out-of-range result */ |
| static inline double retval_errno_erange_overflow(double x) |
| { |
| struct exception exc; |
| exc.arg1 = x; |
| exc.arg2 = x; |
| exc.type = SING; |
| exc.name = (char *)"log1p"; |
| if (_LIB_VERSION == _SVID_) |
| exc.retval = -HUGE; |
| else |
| exc.retval = -infinity_with_flags(AMD_F_DIVBYZERO); |
| if (_LIB_VERSION == _POSIX_) |
| __set_errno(ERANGE); |
| else if (!matherr(&exc)) |
| __set_errno(ERANGE); |
| return exc.retval; |
| } |
| |
| /* Deal with errno for out-of-range argument */ |
| static inline double retval_errno_edom(double x) |
| { |
| struct exception exc; |
| exc.arg1 = x; |
| exc.arg2 = x; |
| exc.type = DOMAIN; |
| exc.name = (char *)"log1p"; |
| if (_LIB_VERSION == _SVID_) |
| exc.retval = -HUGE; |
| else |
| exc.retval = nan_with_flags(AMD_F_INVALID); |
| if (_LIB_VERSION == _POSIX_) |
| __set_errno(EDOM); |
| else if (!matherr(&exc)) |
| { |
| if(_LIB_VERSION == _SVID_) |
| (void)fputs("log1p: DOMAIN error\n", stderr); |
| __set_errno(EDOM); |
| } |
| return exc.retval; |
| } |
| #endif |
| |
| #undef _FUNCNAME |
| #define _FUNCNAME "log1p" |
| |
| double FN_PROTOTYPE(log1p)(double x) |
| { |
| |
| int xexp; |
| double r, r1, r2, correction, f, f1, f2, q, u, v, z1, z2, poly, m2; |
| int index; |
| unsigned long long ux, ax; |
| |
| /* |
| Computes natural log(1+x). Algorithm based on: |
| Ping-Tak Peter Tang |
| "Table-driven implementation of the logarithm function in IEEE |
| floating-point arithmetic" |
| ACM Transactions on Mathematical Software (TOMS) |
| Volume 16, Issue 4 (December 1990) |
| Note that we use a lookup table of size 64 rather than 128, |
| and compensate by having extra terms in the minimax polynomial |
| for the kernel approximation. |
| */ |
| |
| /* Arrays ln_lead_table and ln_tail_table contain |
| leading and trailing parts respectively of precomputed |
| values of natural log(1+i/64), for i = 0, 1, ..., 64. |
| ln_lead_table contains the first 24 bits of precision, |
| and ln_tail_table contains a further 53 bits precision. */ |
| |
| static const double ln_lead_table[65] = { |
| 0.00000000000000000000e+00, /* 0x0000000000000000 */ |
| 1.55041813850402832031e-02, /* 0x3f8fc0a800000000 */ |
| 3.07716131210327148438e-02, /* 0x3f9f829800000000 */ |
| 4.58095073699951171875e-02, /* 0x3fa7745800000000 */ |
| 6.06245994567871093750e-02, /* 0x3faf0a3000000000 */ |
| 7.52233862876892089844e-02, /* 0x3fb341d700000000 */ |
| 8.96121263504028320312e-02, /* 0x3fb6f0d200000000 */ |
| 1.03796780109405517578e-01, /* 0x3fba926d00000000 */ |
| 1.17783010005950927734e-01, /* 0x3fbe270700000000 */ |
| 1.31576299667358398438e-01, /* 0x3fc0d77e00000000 */ |
| 1.45181953907012939453e-01, /* 0x3fc2955280000000 */ |
| 1.58604979515075683594e-01, /* 0x3fc44d2b00000000 */ |
| 1.71850204467773437500e-01, /* 0x3fc5ff3000000000 */ |
| 1.84922337532043457031e-01, /* 0x3fc7ab8900000000 */ |
| 1.97825729846954345703e-01, /* 0x3fc9525a80000000 */ |
| 2.10564732551574707031e-01, /* 0x3fcaf3c900000000 */ |
| 2.23143517971038818359e-01, /* 0x3fcc8ff780000000 */ |
| 2.35566020011901855469e-01, /* 0x3fce270700000000 */ |
| 2.47836112976074218750e-01, /* 0x3fcfb91800000000 */ |
| 2.59957492351531982422e-01, /* 0x3fd0a324c0000000 */ |
| 2.71933674812316894531e-01, /* 0x3fd1675c80000000 */ |
| 2.83768117427825927734e-01, /* 0x3fd22941c0000000 */ |
| 2.95464158058166503906e-01, /* 0x3fd2e8e280000000 */ |
| 3.07025015354156494141e-01, /* 0x3fd3a64c40000000 */ |
| 3.18453729152679443359e-01, /* 0x3fd4618bc0000000 */ |
| 3.29753279685974121094e-01, /* 0x3fd51aad80000000 */ |
| 3.40926527976989746094e-01, /* 0x3fd5d1bd80000000 */ |
| 3.51976394653320312500e-01, /* 0x3fd686c800000000 */ |
| 3.62905442714691162109e-01, /* 0x3fd739d7c0000000 */ |
| 3.73716354370117187500e-01, /* 0x3fd7eaf800000000 */ |
| 3.84411692619323730469e-01, /* 0x3fd89a3380000000 */ |
| 3.94993782043457031250e-01, /* 0x3fd9479400000000 */ |
| 4.05465066432952880859e-01, /* 0x3fd9f323c0000000 */ |
| 4.15827870368957519531e-01, /* 0x3fda9cec80000000 */ |
| 4.26084339618682861328e-01, /* 0x3fdb44f740000000 */ |
| 4.36236739158630371094e-01, /* 0x3fdbeb4d80000000 */ |
| 4.46287095546722412109e-01, /* 0x3fdc8ff7c0000000 */ |
| 4.56237375736236572266e-01, /* 0x3fdd32fe40000000 */ |
| 4.66089725494384765625e-01, /* 0x3fddd46a00000000 */ |
| 4.75845873355865478516e-01, /* 0x3fde744240000000 */ |
| 4.85507786273956298828e-01, /* 0x3fdf128f40000000 */ |
| 4.95077252388000488281e-01, /* 0x3fdfaf5880000000 */ |
| 5.04556000232696533203e-01, /* 0x3fe02552a0000000 */ |
| 5.13945698738098144531e-01, /* 0x3fe0723e40000000 */ |
| 5.23248136043548583984e-01, /* 0x3fe0be72e0000000 */ |
| 5.32464742660522460938e-01, /* 0x3fe109f380000000 */ |
| 5.41597247123718261719e-01, /* 0x3fe154c3c0000000 */ |
| 5.50647079944610595703e-01, /* 0x3fe19ee6a0000000 */ |
| 5.59615731239318847656e-01, /* 0x3fe1e85f40000000 */ |
| 5.68504691123962402344e-01, /* 0x3fe23130c0000000 */ |
| 5.77315330505371093750e-01, /* 0x3fe2795e00000000 */ |
| 5.86049020290374755859e-01, /* 0x3fe2c0e9e0000000 */ |
| 5.94707071781158447266e-01, /* 0x3fe307d720000000 */ |
| 6.03290796279907226562e-01, /* 0x3fe34e2880000000 */ |
| 6.11801505088806152344e-01, /* 0x3fe393e0c0000000 */ |
| 6.20240390300750732422e-01, /* 0x3fe3d90260000000 */ |
| 6.28608644008636474609e-01, /* 0x3fe41d8fe0000000 */ |
| 6.36907458305358886719e-01, /* 0x3fe4618bc0000000 */ |
| 6.45137906074523925781e-01, /* 0x3fe4a4f840000000 */ |
| 6.53301239013671875000e-01, /* 0x3fe4e7d800000000 */ |
| 6.61398470401763916016e-01, /* 0x3fe52a2d20000000 */ |
| 6.69430613517761230469e-01, /* 0x3fe56bf9c0000000 */ |
| 6.77398800849914550781e-01, /* 0x3fe5ad4040000000 */ |
| 6.85303986072540283203e-01, /* 0x3fe5ee02a0000000 */ |
| 6.93147122859954833984e-01}; /* 0x3fe62e42e0000000 */ |
| |
| static const double ln_tail_table[65] = { |
| 0.00000000000000000000e+00, /* 0x0000000000000000 */ |
| 5.15092497094772879206e-09, /* 0x3e361f807c79f3db */ |
| 4.55457209735272790188e-08, /* 0x3e6873c1980267c8 */ |
| 2.86612990859791781788e-08, /* 0x3e5ec65b9f88c69e */ |
| 2.23596477332056055352e-08, /* 0x3e58022c54cc2f99 */ |
| 3.49498983167142274770e-08, /* 0x3e62c37a3a125330 */ |
| 3.23392843005887000414e-08, /* 0x3e615cad69737c93 */ |
| 1.35722380472479366661e-08, /* 0x3e4d256ab1b285e9 */ |
| 2.56504325268044191098e-08, /* 0x3e5b8abcb97a7aa2 */ |
| 5.81213608741512136843e-08, /* 0x3e6f34239659a5dc */ |
| 5.59374849578288093334e-08, /* 0x3e6e07fd48d30177 */ |
| 5.06615629004996189970e-08, /* 0x3e6b32df4799f4f6 */ |
| 5.24588857848400955725e-08, /* 0x3e6c29e4f4f21cf8 */ |
| 9.61968535632653505972e-10, /* 0x3e1086c848df1b59 */ |
| 1.34829655346594463137e-08, /* 0x3e4cf456b4764130 */ |
| 3.65557749306383026498e-08, /* 0x3e63a02ffcb63398 */ |
| 3.33431709374069198903e-08, /* 0x3e61e6a6886b0976 */ |
| 5.13008650536088382197e-08, /* 0x3e6b8abcb97a7aa2 */ |
| 5.09285070380306053751e-08, /* 0x3e6b578f8aa35552 */ |
| 3.20853940845502057341e-08, /* 0x3e6139c871afb9fc */ |
| 4.06713248643004200446e-08, /* 0x3e65d5d30701ce64 */ |
| 5.57028186706125221168e-08, /* 0x3e6de7bcb2d12142 */ |
| 5.48356693724804282546e-08, /* 0x3e6d708e984e1664 */ |
| 1.99407553679345001938e-08, /* 0x3e556945e9c72f36 */ |
| 1.96585517245087232086e-09, /* 0x3e20e2f613e85bda */ |
| 6.68649386072067321503e-09, /* 0x3e3cb7e0b42724f6 */ |
| 5.89936034642113390002e-08, /* 0x3e6fac04e52846c7 */ |
| 2.85038578721554472484e-08, /* 0x3e5e9b14aec442be */ |
| 5.09746772910284482606e-08, /* 0x3e6b5de8034e7126 */ |
| 5.54234668933210171467e-08, /* 0x3e6dc157e1b259d3 */ |
| 6.29100830926604004874e-09, /* 0x3e3b05096ad69c62 */ |
| 2.61974119468563937716e-08, /* 0x3e5c2116faba4cdd */ |
| 4.16752115011186398935e-08, /* 0x3e665fcc25f95b47 */ |
| 2.47747534460820790327e-08, /* 0x3e5a9a08498d4850 */ |
| 5.56922172017964209793e-08, /* 0x3e6de647b1465f77 */ |
| 2.76162876992552906035e-08, /* 0x3e5da71b7bf7861d */ |
| 7.08169709942321478061e-09, /* 0x3e3e6a6886b09760 */ |
| 5.77453510221151779025e-08, /* 0x3e6f0075eab0ef64 */ |
| 4.43021445893361960146e-09, /* 0x3e33071282fb989b */ |
| 3.15140984357495864573e-08, /* 0x3e60eb43c3f1bed2 */ |
| 2.95077445089736670973e-08, /* 0x3e5faf06ecb35c84 */ |
| 1.44098510263167149349e-08, /* 0x3e4ef1e63db35f68 */ |
| 1.05196987538551827693e-08, /* 0x3e469743fb1a71a5 */ |
| 5.23641361722697546261e-08, /* 0x3e6c1cdf404e5796 */ |
| 7.72099925253243069458e-09, /* 0x3e4094aa0ada625e */ |
| 5.62089493829364197156e-08, /* 0x3e6e2d4c96fde3ec */ |
| 3.53090261098577946927e-08, /* 0x3e62f4d5e9a98f34 */ |
| 3.80080516835568242269e-08, /* 0x3e6467c96ecc5cbe */ |
| 5.66961038386146408282e-08, /* 0x3e6e7040d03dec5a */ |
| 4.42287063097349852717e-08, /* 0x3e67bebf4282de36 */ |
| 3.45294525105681104660e-08, /* 0x3e6289b11aeb783f */ |
| 2.47132034530447431509e-08, /* 0x3e5a891d1772f538 */ |
| 3.59655343422487209774e-08, /* 0x3e634f10be1fb591 */ |
| 5.51581770357780862071e-08, /* 0x3e6d9ce1d316eb93 */ |
| 3.60171867511861372793e-08, /* 0x3e63562a19a9c442 */ |
| 1.94511067964296180547e-08, /* 0x3e54e2adf548084c */ |
| 1.54137376631349347838e-08, /* 0x3e508ce55cc8c97a */ |
| 3.93171034490174464173e-09, /* 0x3e30e2f613e85bda */ |
| 5.52990607758839766440e-08, /* 0x3e6db03ebb0227bf */ |
| 3.29990737637586136511e-08, /* 0x3e61b75bb09cb098 */ |
| 1.18436010922446096216e-08, /* 0x3e496f16abb9df22 */ |
| 4.04248680368301346709e-08, /* 0x3e65b3f399411c62 */ |
| 2.27418915900284316293e-08, /* 0x3e586b3e59f65355 */ |
| 1.70263791333409206020e-08, /* 0x3e52482ceae1ac12 */ |
| 5.76999904754328540596e-08}; /* 0x3e6efa39ef35793c */ |
| |
| /* log2_lead and log2_tail sum to an extra-precise version |
| of log(2) */ |
| static const double |
| log2_lead = 6.93147122859954833984e-01, /* 0x3fe62e42e0000000 */ |
| log2_tail = 5.76999904754328540596e-08; /* 0x3e6efa39ef35793c */ |
| |
| static const double |
| /* Approximating polynomial coefficients for x near 0.0 */ |
| ca_1 = 8.33333333333317923934e-02, /* 0x3fb55555555554e6 */ |
| ca_2 = 1.25000000037717509602e-02, /* 0x3f89999999bac6d4 */ |
| ca_3 = 2.23213998791944806202e-03, /* 0x3f62492307f1519f */ |
| ca_4 = 4.34887777707614552256e-04, /* 0x3f3c8034c85dfff0 */ |
| |
| /* Approximating polynomial coefficients for other x */ |
| cb_1 = 8.33333333333333593622e-02, /* 0x3fb5555555555557 */ |
| cb_2 = 1.24999999978138668903e-02, /* 0x3f89999999865ede */ |
| cb_3 = 2.23219810758559851206e-03; /* 0x3f6249423bd94741 */ |
| |
| /* The values exp(-1/16)-1 and exp(1/16)-1 */ |
| static const double |
| log1p_thresh1 = -6.05869371865242201114e-02, /* 0xbfaf0540438fd5c4 */ |
| log1p_thresh2 = 6.44944589178594318568e-02; /* 0x3fb082b577d34ed8 */ |
| |
| |
| GET_BITS_DP64(x, ux); |
| ax = ux & ~SIGNBIT_DP64; |
| |
| if ((ux & EXPBITS_DP64) == EXPBITS_DP64) |
| { |
| /* x is either NaN or infinity */ |
| if (ux & MANTBITS_DP64) |
| { |
| /* x is NaN */ |
| #ifdef WINDOWS |
| return handle_error(_FUNCNAME, ux|0x0008000000000000, _DOMAIN, |
| 0, EDOM, x, 0.0); |
| #else |
| return x + x; /* Raise invalid if it is a signalling NaN */ |
| #endif |
| } |
| else |
| { |
| /* x is infinity */ |
| if (ux & SIGNBIT_DP64) |
| /* x is negative infinity. Return a NaN. */ |
| #ifdef WINDOWS |
| return handle_error(_FUNCNAME, INDEFBITPATT_DP64, _DOMAIN, |
| AMD_F_INVALID, EDOM, x, 0.0); |
| #else |
| return retval_errno_edom(x); |
| #endif |
| else |
| return x; |
| } |
| } |
| else if (ux >= 0xbff0000000000000) |
| { |
| /* x <= -1.0 */ |
| if (ux > 0xbff0000000000000) |
| { |
| /* x is less than -1.0. Return a NaN. */ |
| #ifdef WINDOWS |
| return handle_error(_FUNCNAME, INDEFBITPATT_DP64, _DOMAIN, |
| AMD_F_INVALID, EDOM, x, 0.0); |
| #else |
| return retval_errno_edom(x); |
| #endif |
| } |
| else |
| { |
| /* x is exactly -1.0. Return -infinity with div-by-zero flag. */ |
| #ifdef WINDOWS |
| return handle_error(_FUNCNAME, NINFBITPATT_DP64, _SING, |
| AMD_F_DIVBYZERO, ERANGE, x, 0.0); |
| #else |
| return retval_errno_erange_overflow(x); |
| #endif |
| } |
| } |
| else if (ax < 0x3ca0000000000000) |
| { |
| if (ax == 0x0000000000000000) |
| { |
| /* x is +/-zero. Return the same zero. */ |
| return x; |
| } |
| else |
| /* abs(x) is less than epsilon. Return x with inexact. */ |
| return val_with_flags(x, AMD_F_INEXACT); |
| } |
| |
| |
| if (x < log1p_thresh1 || x > log1p_thresh2) |
| { |
| /* x is outside the range [exp(-1/16)-1, exp(1/16)-1] */ |
| /* |
| First, we decompose the argument x to the form |
| 1 + x = 2**M * (F1 + F2), |
| where 1 <= F1+F2 < 2, M has the value of an integer, |
| F1 = 1 + j/64, j ranges from 0 to 64, and |F2| <= 1/128. |
| |
| Second, we approximate log( 1 + F2/F1 ) by an odd polynomial |
| in U, where U = 2 F2 / (2 F1 + F2). |
| Note that log( 1 + F2/F1 ) = log( 1 + U/2 ) - log( 1 - U/2 ). |
| The core approximation calculates |
| Poly = [log( 1 + U/2 ) - log( 1 - U/2 )]/U - 1. |
| Note that log(1 + U/2) - log(1 - U/2) = 2 arctanh ( U/2 ), |
| thus, Poly = 2 arctanh( U/2 ) / U - 1. |
| |
| It is not hard to see that |
| log(x) = M*log(2) + log(F1) + log( 1 + F2/F1 ). |
| Hence, we return Z1 = log(F1), and Z2 = log( 1 + F2/F1). |
| The values of log(F1) are calculated beforehand and stored |
| in the program. |
| */ |
| |
| f = 1.0 + x; |
| GET_BITS_DP64(f, ux); |
| |
| /* Store the exponent of x in xexp and put |
| f into the range [1.0,2.0) */ |
| xexp = (int)((ux & EXPBITS_DP64) >> EXPSHIFTBITS_DP64) - EXPBIAS_DP64; |
| PUT_BITS_DP64((ux & MANTBITS_DP64) | ONEEXPBITS_DP64, f); |
| |
| /* Now (1+x) = 2**(xexp) * f, 1 <= f < 2. */ |
| |
| /* Set index to be the nearest integer to 64*f */ |
| /* 64 <= index <= 128 */ |
| /* |
| r = 64.0 * f; |
| index = (int)(r + 0.5); |
| */ |
| /* This code instead of the above can save several cycles. |
| It only works because 64 <= r < 128, so |
| the nearest integer is always contained in exactly |
| 7 bits, and the right shift is always the same. */ |
| index = (int)((((ux & 0x000fc00000000000) | 0x0010000000000000) >> 46) |
| + ((ux & 0x0000200000000000) >> 45)); |
| |
| f1 = index * 0.015625; /* 0.015625 = 1/64 */ |
| index -= 64; |
| |
| /* Now take great care to compute f2 such that f1 + f2 = f */ |
| if (xexp <= -2 || xexp >= MANTLENGTH_DP64 + 8) |
| { |
| f2 = f - f1; |
| } |
| else |
| { |
| /* Create the number m2 = 2.0^(-xexp) */ |
| ux = (unsigned long long)(0x3ff - xexp) << EXPSHIFTBITS_DP64; |
| PUT_BITS_DP64(ux,m2); |
| if (xexp <= MANTLENGTH_DP64 - 1) |
| { |
| f2 = (m2 - f1) + m2*x; |
| } |
| else |
| { |
| f2 = (m2*x - f1) + m2; |
| } |
| } |
| |
| /* At this point, x = 2**xexp * ( f1 + f2 ) where |
| f1 = j/64, j = 1, 2, ..., 64 and |f2| <= 1/128. */ |
| |
| z1 = ln_lead_table[index]; |
| q = ln_tail_table[index]; |
| |
| /* Calculate u = 2 f2 / ( 2 f1 + f2 ) = f2 / ( f1 + 0.5*f2 ) */ |
| u = f2 / (f1 + 0.5 * f2); |
| |
| /* Here, |u| <= 2(exp(1/16)-1) / (exp(1/16)+1). |
| The core approximation calculates |
| poly = [log(1 + u/2) - log(1 - u/2)]/u - 1 */ |
| v = u * u; |
| poly = (v * (cb_1 + v * (cb_2 + v * cb_3))); |
| z2 = q + (u + u * poly); |
| |
| /* Now z1,z2 is an extra-precise approximation of log(f). */ |
| |
| /* Add xexp * log(2) to z1,z2 to get the result log(1+x). |
| The computed r1 is not subject to rounding error because |
| xexp has at most 10 significant bits, log(2) has 24 significant |
| bits, and z1 has up to 24 bits; and the exponents of z1 |
| and z2 differ by at most 6. */ |
| r1 = (xexp * log2_lead + z1); |
| r2 = (xexp * log2_tail + z2); |
| /* Natural log(1+x) */ |
| return r1 + r2; |
| } |
| else |
| { |
| /* Arguments close to 0.0 are handled separately to maintain |
| accuracy. |
| |
| The approximation in this region exploits the identity |
| log( 1 + r ) = log( 1 + u/2 ) - log( 1 - u/2 ), where |
| u = 2r / (2+r). |
| Note that the right hand side has an odd Taylor series expansion |
| which converges much faster than the Taylor series expansion of |
| log( 1 + r ) in r. Thus, we approximate log( 1 + r ) by |
| u + A1 * u^3 + A2 * u^5 + ... + An * u^(2n+1). |
| |
| One subtlety is that since u cannot be calculated from |
| r exactly, the rounding error in the first u should be |
| avoided if possible. To accomplish this, we observe that |
| u = r - r*r/(2+r). |
| Since x (=r) is the input argument, and thus presumed exact, |
| the formula above approximates u accurately because |
| u = r - correction, |
| and the magnitude of "correction" (of the order of r*r) |
| is small. |
| With these observations, we will approximate log( 1 + r ) by |
| r + ( (A1*u^3 + ... + An*u^(2n+1)) - correction ). |
| |
| We approximate log(1+r) by an odd polynomial in u, where |
| u = 2r/(2+r) = r - r*r/(2+r). |
| */ |
| r = x; |
| u = r / (2.0 + r); |
| correction = r * u; |
| u = u + u; |
| v = u * u; |
| r1 = r; |
| r2 = (u * v * (ca_1 + v * (ca_2 + v * (ca_3 + v * ca_4))) - correction); |
| return r1 + r2; |
| } |
| } |
| |
| weak_alias (__log1p, log1p) |
