/* ============================================================ Copyright (c) 2002-2015 Advanced Micro Devices, Inc. All rights reserved. Redistribution and use in source and binary forms, with or without modification, are permitted provided that the following conditions are met: + Redistributions of source code must retain the above copyright notice, this list of conditions and the following disclaimer. + Redistributions in binary form must reproduce the above copyright notice, this list of conditions and the following disclaimer in the documentation and/or other materials provided with the distribution. + Neither the name of Advanced Micro Devices, Inc. nor the names of its contributors may 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 ADVANCED MICRO DEVICES, INC. OR 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. It is licensee's responsibility to comply with any export regulations applicable in licensee's jurisdiction. ============================================================ */ #include "libm_amd.h" #include "libm_util_amd.h" #define USE_NANF_WITH_FLAGS #define USE_INFINITYF_WITH_FLAGS #define USE_HANDLE_ERRORF #include "libm_inlines_amd.h" #undef USE_NANF_WITH_FLAGS #undef USE_INFINITYF_WITH_FLAGS #undef USE_HANDLE_ERRORF #include "libm_errno_amd.h" /* Deal with errno for out-of-range result */ static inline float retval_errno_erange_overflow(float x __attribute__((unused))) { return -infinityf_with_flags(AMD_F_DIVBYZERO); } /* Deal with errno for out-of-range argument */ static inline float retval_errno_edom(float x __attribute__((unused))) { return nanf_with_flags(AMD_F_INVALID); } #undef _FUNCNAME #if defined(COMPILING_LOG10) #define _FUNCNAME "log10f" float FN_PROTOTYPE(mth_i_log10)(float fx) #elif defined(COMPILING_LOG2) #define _FUNCNAME "log2f" float FN_PROTOTYPE(mth_i_log2)(float fx) #else #define _FUNCNAME "logf" float FN_PROTOTYPE(mth_i_log)(float fx) #endif { double x = fx; int xexp; double r, f, f1, f2, q, u, v, z1, z2, poly; int index; __UINT8_T ux; /* Computes natural log(x) for float arguments. Algorithm is basically a promotion of the arguments to double followed by an inlined version of the double algorithm, simplified for efficiency (see log_amd.c). Simplifications include: * Special algorithm for arguments near 1.0 not required * Scaling of denormalised arguments not required * Shorter core series approximations used */ /* 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 */ static const double log2 = 6.931471805599453e-01, /* 0x3fe62e42fefa39ef */ /* Approximating polynomial coefficients */ cb_1 = 8.33333333333333593622e-02, /* 0x3fb5555555555557 */ cb_2 = 1.24999999978138668903e-02; /* 0x3f89999999865ede */ #if defined(COMPILING_LOG10) static const double log10e = 4.34294481903251827651e-01; /* 0x3fdbcb7b1526e50e */ #elif defined(COMPILING_LOG2) static const double log2e = 1.44269504088896340735e+00; /* 0x3ff71547652b82fe */ #endif GET_BITS_DP64(x, ux); #if !defined(COMPILING_LOG10) && !defined(COMPILING_LOG2) if (ux == 0x4005bf0a80000000) /* Treat this, the number closest to e in float arithmetic, as a special case and return 1.0 */ return 1.0F; #endif if ((ux & EXPBITS_DP64) == EXPBITS_DP64) { /* x is either NaN or infinity */ if (ux & MANTBITS_DP64) { /* x is NaN */ return fx + fx; /* Raise invalid if it is a signalling NaN */ } else { /* x is infinity */ if (ux & SIGNBIT_DP64) { /* x is negative infinity. Return a NaN. */ return retval_errno_edom(fx); } else return fx; } } else if (!(ux & ~SIGNBIT_DP64)) { /* x is +/-zero. Return -infinity with div-by-zero flag. */ return retval_errno_erange_overflow(fx); } else if (ux & SIGNBIT_DP64) { /* x is negative. Return a NaN. */ return retval_errno_edom(fx); } /* First, we decompose the argument x to the form 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 F2 + F1). 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 = x; /* Store the exponent of x in xexp and put f into the range [0.5,1) */ xexp = (int)((ux & EXPBITS_DP64) >> EXPSHIFTBITS_DP64) - EXPBIAS_DP64; PUT_BITS_DP64((ux & MANTBITS_DP64) | HALFEXPBITS_DP64, f); /* Now x = 2**xexp * f, 1/2 <= f < 1. */ /* Set index to be the nearest integer to 128*f */ /* r = 128.0 * 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)); z1 = ln_lead_table[index - 64]; q = ln_tail_table[index - 64]; f1 = index * 0.0078125; /* 0.0078125 = 1/128 */ f2 = f - f1; /* At this point, x = 2**xexp * ( f1 + f2 ) where f1 = j/128, j = 64, 65, ..., 128 and |f2| <= 1/256. */ /* Calculate u = 2 f2 / ( 2 f1 + f2 ) = f2 / ( f1 + 0.5*f2 ) */ /* u = 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)); 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(x). */ r = xexp * log2 + z1 + z2; #if defined(COMPILING_LOG10) return (float)(log10e * r); #elif defined(COMPILING_LOG2) return (float)(log2e * r); #else return (float)r; #endif }