.file "exp2f.s" // Copyright (c) 2000 - 2005, Intel Corporation // All rights reserved. // // Contributed 2000 by the Intel Numerics Group, Intel Corporation // // Redistribution and use in source and binary forms, with or without // modification, are permitted provided that the following conditions are // met: // // * Redistributions of source code must retain the above copyright // notice, this list of conditions and the following disclaimer. // // * Redistributions in binary form must reproduce the above copyright // notice, this list of conditions and the following disclaimer in the // documentation and/or other materials provided with the distribution. // // * The name of Intel Corporation may not be used to endorse or promote // products derived from this software without specific prior written // permission. // THIS SOFTWARE IS PROVIDED BY THE COPYRIGHT HOLDERS AND CONTRIBUTORS // "AS IS" AND ANY EXPRESS OR IMPLIED WARRANTIES, INCLUDING, BUT NOT // LIMITED TO, THE IMPLIED WARRANTIES OF MERCHANTABILITY AND FITNESS FOR // A PARTICULAR PURPOSE ARE DISCLAIMED. IN NO EVENT SHALL INTEL OR ITS // CONTRIBUTORS BE LIABLE FOR ANY DIRECT, INDIRECT, INCIDENTAL, SPECIAL, // EXEMPLARY, OR CONSEQUENTIAL DAMAGES (INCLUDING, BUT NOT LIMITED TO, // PROCUREMENT OF SUBSTITUTE GOODS OR SERVICES; LOSS OF USE, DATA, OR // PROFITS; OR BUSINESS INTERRUPTION) HOWEVER CAUSED AND ON ANY THEORY // OF LIABILITY, WHETHER IN CONTRACT, STRICT LIABILITY OR TORT (INCLUDING // NEGLIGENCE OR OTHERWISE) ARISING IN ANY WAY OUT OF THE USE OF THIS // SOFTWARE, EVEN IF ADVISED OF THE POSSIBILITY OF SUCH DAMAGE. // // Intel Corporation is the author of this code, and requests that all // problem reports or change requests be submitted to it directly at // http://www.intel.com/software/products/opensource/libraries/num.htm. // // History //============================================================== // 08/25/00 Initial version // 05/20/02 Cleaned up namespace and sf0 syntax // 09/05/02 Improved performance and accuracy // 01/17/03 Fixed to call error support when x=128.0 // 03/31/05 Reformatted delimiters between data tables // // API //============================================================== // float exp2f(float) // // Overview of operation //============================================================== // Background // // Implementation // // Let x= (K + fh + fl + r), where // K is an integer, fh= 0.b1 b2 b3 b4 b5, // fl= 2^{-5}* 0.b6 b7 b8 b8 b10 (fh, fl >= 0), // and |r|<2^{-11} // Th is a table that stores 2^fh (32 entries) rounded to // double extended precision (only mantissa is stored) // Tl is a table that stores 2^fl (32 entries) rounded to // double extended precision (only mantissa is stored) // // 2^x is approximated as // 2^K * Th [ f ] * Tl [ f ] * (1+c1*r+c2*r^2) // Note: We use the following trick to speed up conversion from FP to integer: // // Let x = K + r, where K is an integer, and |r| <= 0.5 // Let N be the number of significand bits for the FP format used // ( N=64 for double-extended, N=53 for double) // // Then let y = 1.5 * 2^(N-1) + x for RN mode // K = y - 1.5 * 2^(N-1) // r = x - K // // If we want to obtain the integer part and the first m fractional bits of x, // we can use the same trick, but with a constant of 1.5 * 2^(N-1-m): // // Let x = K + f + r // f = 0.b_1 b_2 ... b_m // |r| <= 2^(-m-1) // // Then let y = 1.5 * 2^(N-1-m) + x for RN mode // (K+f) = y - 1.5 * 2^(N-1-m) // r = x - K // Special values //============================================================== // exp2(0)= 1 // exp2(+inf)= inf // exp2(-inf)= 0 // // Registers used //============================================================== // r2-r3, r14-r40 // f6-f15, f32-f45 // p6-p8, p12 // GR_TBL_START = r2 GR_LOG_TBL = r3 GR_OF_LIMIT = r14 GR_UF_LIMIT = r15 GR_EXP_CORR = r16 GR_F_low = r17 GR_F_high = r18 GR_K = r19 GR_Flow_ADDR = r20 GR_BIAS = r21 GR_Fh = r22 GR_Fh_ADDR = r23 GR_EXPMAX = r24 GR_EMIN = r25 GR_ROUNDVAL = r26 GR_MASK = r27 GR_KF0 = r28 GR_MASK_low = r29 GR_COEFF_START = r30 GR_SAVE_B0 = r33 GR_SAVE_PFS = r34 GR_SAVE_GP = r35 GR_SAVE_SP = r36 GR_Parameter_X = r37 GR_Parameter_Y = r38 GR_Parameter_RESULT = r39 GR_Parameter_TAG = r40 FR_X = f10 FR_Y = f1 FR_RESULT = f8 FR_COEFF1 = f6 FR_COEFF2 = f7 FR_R = f9 FR_KF0 = f12 FR_UF_LIMIT = f15 FR_OF_LIMIT = f32 FR_EXPMIN = f33 FR_ROUNDVAL = f34 FR_KF = f35 FR_2_TO_K = f36 FR_T_low = f37 FR_T_high = f38 FR_P12 = f41 FR_T_low_K = f42 FR_T = f44 FR_P = f45 // Data tables //============================================================== RODATA .align 16 LOCAL_OBJECT_START(poly_coeffs) data8 0xb17217f7d1cf79ab, 0x00003ffe // C_1 data8 0xf5fdeffc162c7541, 0x00003ffc // C_2 LOCAL_OBJECT_END(poly_coeffs) LOCAL_OBJECT_START(T_table) // 2^{0.00000 b6 b7 b8 b9 b10} data8 0x8000000000000000, 0x8016302f17467628 data8 0x802c6436d0e04f50, 0x80429c17d77c18ed data8 0x8058d7d2d5e5f6b0, 0x806f17687707a7af data8 0x80855ad965e88b83, 0x809ba2264dada76a data8 0x80b1ed4fd999ab6c, 0x80c83c56b50cf77f data8 0x80de8f3b8b85a0af, 0x80f4e5ff089f763e data8 0x810b40a1d81406d4, 0x81219f24a5baa59d data8 0x813801881d886f7b, 0x814e67cceb90502c data8 0x8164d1f3bc030773, 0x817b3ffd3b2f2e47 data8 0x8191b1ea15813bfd, 0x81a827baf7838b78 data8 0x81bea1708dde6055, 0x81d51f0b8557ec1c data8 0x81eba08c8ad4536f, 0x820225f44b55b33b data8 0x8218af4373fc25eb, 0x822f3c7ab205c89a data8 0x8245cd9ab2cec048, 0x825c62a423d13f0c data8 0x8272fb97b2a5894c, 0x828998760d01faf3 data8 0x82a0393fe0bb0ca8, 0x82b6ddf5dbc35906 // // 2^{0.b1 b2 b3 b4 b5} data8 0x8000000000000000, 0x82cd8698ac2ba1d7 data8 0x85aac367cc487b14, 0x88980e8092da8527 data8 0x8b95c1e3ea8bd6e6, 0x8ea4398b45cd53c0 data8 0x91c3d373ab11c336, 0x94f4efa8fef70961 data8 0x9837f0518db8a96f, 0x9b8d39b9d54e5538 data8 0x9ef5326091a111ad, 0xa27043030c496818 data8 0xa5fed6a9b15138ea, 0xa9a15ab4ea7c0ef8 data8 0xad583eea42a14ac6, 0xb123f581d2ac258f data8 0xb504f333f9de6484, 0xb8fbaf4762fb9ee9 data8 0xbd08a39f580c36be, 0xc12c4cca66709456 data8 0xc5672a115506dadd, 0xc9b9bd866e2f27a2 data8 0xce248c151f8480e3, 0xd2a81d91f12ae45a data8 0xd744fccad69d6af4, 0xdbfbb797daf23755 data8 0xe0ccdeec2a94e111, 0xe5b906e77c8348a8 data8 0xeac0c6e7dd24392e, 0xefe4b99bdcdaf5cb data8 0xf5257d152486cc2c, 0xfa83b2db722a033a LOCAL_OBJECT_END(T_table) .section .text GLOBAL_LIBM_ENTRY(exp2f) {.mfi alloc r32= ar.pfs, 1, 4, 4, 0 // will continue only for non-zero normal/denormal numbers fclass.nm p12, p0= f8, 0x1b // GR_TBL_START= pointer to C_1...C_2 followed by T_table addl GR_TBL_START= @ltoff(poly_coeffs), gp } {.mlx mov GR_OF_LIMIT= 0xffff + 7 // Exponent of overflow limit movl GR_ROUNDVAL= 0x5a400000 // 1.5*2^(63-10) (SP) } ;; // Form special constant 1.5*2^(63-10) to give integer part and first 10 // fractional bits of x {.mfi setf.s FR_ROUNDVAL= GR_ROUNDVAL // Form special constant fcmp.lt.s1 p6, p8= f8, f0 // X<0 ? nop.i 0 } {.mfb ld8 GR_COEFF_START= [ GR_TBL_START ] // Load pointer to coeff table nop.f 0 (p12) br.cond.spnt SPECIAL_exp2 // Branch if nan, inf, zero } ;; {.mlx setf.exp FR_OF_LIMIT= GR_OF_LIMIT // Set overflow limit movl GR_UF_LIMIT= 0xc3160000 // (-2^7-22) = -150 } ;; {.mfi ldfe FR_COEFF1= [ GR_COEFF_START ], 16 // load C_1 fma.s0 f8= f8, f1, f0 // normalize x nop.i 0 } ;; {.mmi ldfe FR_COEFF2= [ GR_COEFF_START ], 16 // load C_2 setf.s FR_UF_LIMIT= GR_UF_LIMIT // Set underflow limit mov GR_EXP_CORR= 0xffff-126 } ;; {.mfi nop.m 0 fma.s1 FR_KF0= f8, f1, FR_ROUNDVAL // y= x + 1.5*2^(63-10) nop.i 0 } ;; {.mfi mov GR_MASK= 1023 fms.s1 FR_KF= FR_KF0, f1, FR_ROUNDVAL // (K+f) mov GR_MASK_low= 31 } ;; {.mfi getf.sig GR_KF0= FR_KF0 // (K+f)*2^10= round_to_int(y) fcmp.ge.s1 p12, p7= f8, FR_OF_LIMIT // x >= overflow threshold ? add GR_LOG_TBL= 256, GR_COEFF_START // Pointer to high T_table } ;; {.mmi and GR_F_low= GR_KF0, GR_MASK_low // f_low and GR_F_high= GR_MASK, GR_KF0 // f_high*32 shr GR_K= GR_KF0, 10 // K } ;; {.mmi shladd GR_Flow_ADDR= GR_F_low, 3, GR_COEFF_START // address of 2^{f_low} add GR_BIAS= GR_K, GR_EXP_CORR // K= bias-2*63 shr GR_Fh= GR_F_high, 5 // f_high } ;; {.mfi setf.exp FR_2_TO_K= GR_BIAS // 2^{K-126} fnma.s1 FR_R= FR_KF, f1, f8 // r= x - (K+f) shladd GR_Fh_ADDR= GR_Fh, 3, GR_LOG_TBL // address of 2^{f_high} } {.mlx ldf8 FR_T_low= [ GR_Flow_ADDR ] // load T_low= 2^{f_low} movl GR_EMIN= 0xc2fc0000 // EMIN= -126 } ;; {.mfi ldf8 FR_T_high= [ GR_Fh_ADDR ] // load T_high= 2^{f_high} (p7) fcmp.lt.s1 p12, p7= f8, FR_UF_LIMIT // x