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1//===----------------------------------------------------------------------===//2//3// Part of the LLVM Project, under the Apache License v2.0 with LLVM Exceptions.4// See https://llvm.org/LICENSE.txt for license information.5// SPDX-License-Identifier: Apache-2.0 WITH LLVM-exception6//7//===----------------------------------------------------------------------===//8 9#include <clc/math/clc_fabs.h>10#include <clc/math/clc_fma.h>11#include <clc/math/clc_mad.h>12#include <clc/math/math.h>13#include <clc/relational/clc_isinf.h>14#include <clc/relational/clc_isnan.h>15 16/*17 Algorithm:18 19 Based on:20 Ping-Tak Peter Tang21 "Table-driven implementation of the logarithm function in IEEE22 floating-point arithmetic"23 ACM Transactions on Mathematical Software (TOMS)24 Volume 16, Issue 4 (December 1990)25 26 27 x very close to 1.0 is handled differently, for x everywhere else28 a brief explanation is given below29 30 x = (2^m)*A31 x = (2^m)*(G+g) with (1 <= G < 2) and (g <= 2^(-8))32 x = (2^m)*2*(G/2+g/2)33 x = (2^m)*2*(F+f) with (0.5 <= F < 1) and (f <= 2^(-9))34 35 Y = (2^(-1))*(2^(-m))*(2^m)*A36 Now, range of Y is: 0.5 <= Y < 137 38 F = 0x80 + (first 7 mantissa bits) + (8th mantissa bit)39 Now, range of F is: 128 <= F <= 25640 F = F / 25641 Now, range of F is: 0.5 <= F <= 142 43 f = -(Y-F), with (f <= 2^(-9))44 45 log(x) = m*log(2) + log(2) + log(F-f)46 log(x) = m*log(2) + log(2) + log(F) + log(1-(f/F))47 log(x) = m*log(2) + log(2*F) + log(1-r)48 49 r = (f/F), with (r <= 2^(-8))50 r = f*(1/F) with (1/F) precomputed to avoid division51 52 log(x) = m*log(2) + log(G) - poly53 54 log(G) is precomputed55 poly = (r + (r^2)/2 + (r^3)/3 + (r^4)/4) + (r^5)/5))56 57 log(2) and log(G) need to be maintained in extra precision58 to avoid losing precision in the calculations59 60 61 For x close to 1.0, we employ the following technique to62 ensure faster convergence.63 64 log(x) = log((1+s)/(1-s)) = 2*s + (2/3)*s^3 + (2/5)*s^5 + (2/7)*s^765 x = ((1+s)/(1-s))66 x = 1 + r67 s = r/(2+r)68 69*/70 71_CLC_OVERLOAD _CLC_DEF float72#if defined(COMPILING_LOG2)73__clc_log2(float x)74#elif defined(COMPILING_LOG10)75__clc_log10(float x)76#else77__clc_log(float x)78#endif79{80 81#if defined(COMPILING_LOG2)82 const float LOG2E = 0x1.715476p+0f; // 1.442695040888963483 const float LOG2E_HEAD = 0x1.700000p+0f; // 1.437584 const float LOG2E_TAIL = 0x1.547652p-8f; // 0.0051950407285#elif defined(COMPILING_LOG10)86 const float LOG10E = 0x1.bcb7b2p-2f; // 0.4342944819032518287 const float LOG10E_HEAD = 0x1.bc0000p-2f; // 0.4335937588 const float LOG10E_TAIL = 0x1.6f62a4p-11f; // 0.000700731989 const float LOG10_2_HEAD = 0x1.340000p-2f; // 0.3007812590 const float LOG10_2_TAIL = 0x1.04d426p-12f; // 0.00024874563791#else92 const float LOG2_HEAD = 0x1.62e000p-1f; // 0.69311523493 const float LOG2_TAIL = 0x1.0bfbe8p-15f; // 0.000031946183394#endif95 96 uint xi = __clc_as_uint(x);97 uint ax = xi & EXSIGNBIT_SP32;98 99 // Calculations for |x-1| < 2^-4100 float r = x - 1.0f;101 int near1 = __clc_fabs(r) < 0x1.0p-4f;102 float u2 = MATH_DIVIDE(r, 2.0f + r);103 float corr = u2 * r;104 float u = u2 + u2;105 float v = u * u;106 float znear1, z1, z2;107 108 // 2/(5 * 2^5), 2/(3 * 2^3)109 z2 = __clc_mad(u, __clc_mad(v, 0x1.99999ap-7f, 0x1.555556p-4f) * v, -corr);110 111#if defined(COMPILING_LOG2)112 z1 = __clc_as_float(__clc_as_int(r) & 0xffff0000);113 z2 = z2 + (r - z1);114 znear1 = __clc_mad(115 z1, LOG2E_HEAD,116 __clc_mad(z2, LOG2E_HEAD, __clc_mad(z1, LOG2E_TAIL, z2 * LOG2E_TAIL)));117#elif defined(COMPILING_LOG10)118 z1 = __clc_as_float(__clc_as_int(r) & 0xffff0000);119 z2 = z2 + (r - z1);120 znear1 = __clc_mad(121 z1, LOG10E_HEAD,122 __clc_mad(z2, LOG10E_HEAD, __clc_mad(z1, LOG10E_TAIL, z2 * LOG10E_TAIL)));123#else124 znear1 = z2 + r;125#endif126 127 // Calculations for x not near 1128 int m = (int)(xi >> EXPSHIFTBITS_SP32) - EXPBIAS_SP32;129 130 // Normalize subnormal131 uint xis = __clc_as_uint(__clc_as_float(xi | 0x3f800000) - 1.0f);132 int ms = (int)(xis >> EXPSHIFTBITS_SP32) - 253;133 int c = m == -127;134 m = c ? ms : m;135 uint xin = c ? xis : xi;136 137 float mf = (float)m;138 uint indx = (xin & 0x007f0000) + ((xin & 0x00008000) << 1);139 140 // F - Y141 float f = __clc_as_float(0x3f000000 | indx) -142 __clc_as_float(0x3f000000 | (xin & MANTBITS_SP32));143 144 indx = indx >> 16;145 r = f * __CLC_USE_TABLE(log_inv_tbl, indx);146 147 // 1/3, 1/2148 float poly = __clc_mad(__clc_mad(r, 0x1.555556p-2f, 0.5f), r * r, r);149 150#if defined(COMPILING_LOG2)151 float2 tv = __CLC_USE_TABLE(log2_tbl, indx);152 z1 = tv.s0 + mf;153 z2 = __clc_mad(poly, -LOG2E, tv.s1);154#elif defined(COMPILING_LOG10)155 float2 tv = __CLC_USE_TABLE(log10_tbl, indx);156 z1 = __clc_mad(mf, LOG10_2_HEAD, tv.s0);157 z2 = __clc_mad(poly, -LOG10E, mf * LOG10_2_TAIL) + tv.s1;158#else159 float2 tv = __CLC_USE_TABLE(log_tbl, indx);160 z1 = __clc_mad(mf, LOG2_HEAD, tv.s0);161 z2 = __clc_mad(mf, LOG2_TAIL, -poly) + tv.s1;162#endif163 164 float z = z1 + z2;165 z = near1 ? znear1 : z;166 167 // Corner cases168 z = ax >= PINFBITPATT_SP32 ? x : z;169 z = xi != ax ? __clc_as_float(QNANBITPATT_SP32) : z;170 z = ax == 0 ? __clc_as_float(NINFBITPATT_SP32) : z;171 172 return z;173}174 175#ifdef cl_khr_fp64176 177_CLC_OVERLOAD _CLC_DEF double178#if defined(COMPILING_LOG2)179__clc_log2(double x)180#elif defined(COMPILING_LOG10)181__clc_log10(double x)182#else183__clc_log(double x)184#endif185{186 187#ifndef COMPILING_LOG2188 // log2_lead and log2_tail sum to an extra-precise version of ln(2)189 const double log2_lead = 6.93147122859954833984e-01; /* 0x3fe62e42e0000000 */190 const double log2_tail = 5.76999904754328540596e-08; /* 0x3e6efa39ef35793c */191#endif192 193#if defined(COMPILING_LOG10)194 // log10e_lead and log10e_tail sum to an extra-precision version of log10(e)195 // (19 bits in lead)196 const double log10e_lead =197 4.34293746948242187500e-01; /* 0x3fdbcb7800000000 */198 const double log10e_tail =199 7.3495500964015109100644e-7; /* 0x3ea8a93728719535 */200#elif defined(COMPILING_LOG2)201 // log2e_lead and log2e_tail sum to an extra-precision version of log2(e) (19202 // bits in lead)203 const double log2e_lead = 1.44269180297851562500E+00; /* 0x3FF7154400000000 */204 const double log2e_tail = 3.23791044778235969970E-06; /* 0x3ECB295C17F0BBBE */205#endif206 207 // log_thresh1 = 9.39412117004394531250e-1 = 0x3fee0faa00000000208 // log_thresh2 = 1.06449508666992187500 = 0x3ff1082c00000000209 const double log_thresh1 = 0x1.e0faap-1;210 const double log_thresh2 = 0x1.1082cp+0;211 212 bool is_near = x >= log_thresh1 && x <= log_thresh2;213 214 // Near 1 code215 double r = x - 1.0;216 double u = r / (2.0 + r);217 double correction = r * u;218 u = u + u;219 double v = u * u;220 double r1 = r;221 222 const double ca_1 = 8.33333333333317923934e-02; /* 0x3fb55555555554e6 */223 const double ca_2 = 1.25000000037717509602e-02; /* 0x3f89999999bac6d4 */224 const double ca_3 = 2.23213998791944806202e-03; /* 0x3f62492307f1519f */225 const double ca_4 = 4.34887777707614552256e-04; /* 0x3f3c8034c85dfff0 */226 227 double r2 = __clc_fma(228 u * v, __clc_fma(v, __clc_fma(v, __clc_fma(v, ca_4, ca_3), ca_2), ca_1),229 -correction);230 231#if defined(COMPILING_LOG10)232 r = r1;233 r1 = __clc_as_double(__clc_as_ulong(r1) & 0xffffffff00000000);234 r2 = r2 + (r - r1);235 double ret_near = __clc_fma(236 log10e_lead, r1,237 __clc_fma(log10e_lead, r2, __clc_fma(log10e_tail, r1, log10e_tail * r2)));238#elif defined(COMPILING_LOG2)239 r = r1;240 r1 = __clc_as_double(__clc_as_ulong(r1) & 0xffffffff00000000);241 r2 = r2 + (r - r1);242 double ret_near = __clc_fma(243 log2e_lead, r1,244 __clc_fma(log2e_lead, r2, __clc_fma(log2e_tail, r1, log2e_tail * r2)));245#else246 double ret_near = r1 + r2;247#endif248 249 // This is the far from 1 code250 251 // Deal with subnormal252 ulong ux = __clc_as_ulong(x);253 ulong uxs =254 __clc_as_ulong(__clc_as_double(0x03d0000000000000UL | ux) - 0x1.0p-962);255 int c = ux < IMPBIT_DP64;256 ux = c ? uxs : ux;257 int expadjust = c ? 60 : 0;258 259 int xexp = ((__clc_as_int2(ux).hi >> 20) & 0x7ff) - EXPBIAS_DP64 - expadjust;260 double f = __clc_as_double(HALFEXPBITS_DP64 | (ux & MANTBITS_DP64));261 int index = __clc_as_int2(ux).hi >> 13;262 index = ((0x80 | (index & 0x7e)) >> 1) + (index & 0x1);263 264 double z1 = __CLC_USE_TABLE(ln_tbl_lo, index - 64);265 double q = __CLC_USE_TABLE(ln_tbl_hi, index - 64);266 267 double f1 = index * 0x1.0p-7;268 double f2 = f - f1;269 u = f2 / __clc_fma(f2, 0.5, f1);270 v = u * u;271 272 const double cb_1 = 8.33333333333333593622e-02; /* 0x3fb5555555555557 */273 const double cb_2 = 1.24999999978138668903e-02; /* 0x3f89999999865ede */274 const double cb_3 = 2.23219810758559851206e-03; /* 0x3f6249423bd94741 */275 276 double poly = v * __clc_fma(v, __clc_fma(v, cb_3, cb_2), cb_1);277 double z2 = q + __clc_fma(u, poly, u);278 279 double dxexp = (double)xexp;280#if defined(COMPILING_LOG10)281 // Add xexp * log(2) to z1,z2 to get log(x)282 r1 = __clc_fma(dxexp, log2_lead, z1);283 r2 = __clc_fma(dxexp, log2_tail, z2);284 double ret_far = __clc_fma(285 log10e_lead, r1,286 __clc_fma(log10e_lead, r2, __clc_fma(log10e_tail, r1, log10e_tail * r2)));287#elif defined(COMPILING_LOG2)288 r1 = __clc_fma(log2e_lead, z1, dxexp);289 r2 = __clc_fma(log2e_lead, z2, __clc_fma(log2e_tail, z1, log2e_tail * z2));290 double ret_far = r1 + r2;291#else292 r1 = __clc_fma(dxexp, log2_lead, z1);293 r2 = __clc_fma(dxexp, log2_tail, z2);294 double ret_far = r1 + r2;295#endif296 297 double ret = is_near ? ret_near : ret_far;298 299 ret = __clc_isinf(x) ? __clc_as_double(PINFBITPATT_DP64) : ret;300 ret = (__clc_isnan(x) | (x < 0.0)) ? __clc_as_double(QNANBITPATT_DP64) : ret;301 ret = x == 0.0 ? __clc_as_double(NINFBITPATT_DP64) : ret;302 return ret;303}304 305#endif // cl_khr_fp64306 307#ifdef cl_khr_fp16308 309_CLC_OVERLOAD _CLC_DEF half310#if defined(COMPILING_LOG2)311__clc_log2(half x) {312 return (half)__clc_log2((float)x);313}314#elif defined(COMPILING_LOG10)315__clc_log10(half x) {316 return (half)__clc_log10((float)x);317}318#else319__clc_log(half x) {320 return (half)__clc_log((float)x);321}322#endif323 324#endif // cl_khr_fp16325