209 changes: 15 additions & 194 deletions libc/src/math/generic/exp.cpp
View file
Original file line number Diff line number Diff line change
Expand Up @@ -8,6 +8,7 @@

#include "src/math/exp.h"
#include "common_constants.h" // Lookup tables EXP_M1 and EXP_M2.
#include "explogxf.h" // ziv_test_denorm.
#include "src/__support/CPP/bit.h"
#include "src/__support/CPP/optional.h"
#include "src/__support/FPUtil/FEnvImpl.h"
Expand All @@ -18,6 +19,7 @@
#include "src/__support/FPUtil/multiply_add.h"
#include "src/__support/FPUtil/nearest_integer.h"
#include "src/__support/FPUtil/rounding_mode.h"
#include "src/__support/FPUtil/triple_double.h"
#include "src/__support/common.h"
#include "src/__support/macros/optimization.h" // LIBC_UNLIKELY

Expand All @@ -26,9 +28,10 @@
namespace __llvm_libc {

using fputil::DoubleDouble;
using fputil::TripleDouble;
using Float128 = typename fputil::DyadicFloat<128>;

// 2^12 * log2(e)
// log2(e)
constexpr double LOG2_E = 0x1.71547652b82fep+0;

// Error bounds:
Expand All @@ -37,12 +40,6 @@ constexpr double ERR_D = 0x1.8p-63;
// Errors when using double-double precision.
constexpr double ERR_DD = 0x1.0p-99;

struct TripleDouble {
double hi = 0.0;
double mid = 0.0;
double lo = 0.0;
};

// -2^-12 * log(2)
// > a = -2^-12 * log(2);
// > b = round(a, 30, RN);
Expand All @@ -54,142 +51,6 @@ constexpr double MLOG_2_EXP2_M12_MID = 0x1.718432a1b0e26p-47;
constexpr double MLOG_2_EXP2_M12_MID_30 = 0x1.718432ap-47;
constexpr double MLOG_2_EXP2_M12_LO = 0x1.b0e2633fe0685p-79;

// 2^(k * 2^-6), for k = 0..63.
constexpr TripleDouble EXP_MID1[64] = {
{0x1p0, 0, 0},
{0x1.02c9a3e778061p0, -0x1.19083535b085dp-56, -0x1.9085b0a3d74d5p-110},
{0x1.059b0d3158574p0, 0x1.d73e2a475b465p-55, 0x1.05ff94f8d257ep-110},
{0x1.0874518759bc8p0, 0x1.186be4bb284ffp-57, 0x1.15820d96b414fp-111},
{0x1.0b5586cf9890fp0, 0x1.8a62e4adc610bp-54, -0x1.67c9bd6ebf74cp-108},
{0x1.0e3ec32d3d1a2p0, 0x1.03a1727c57b53p-59, -0x1.5aa76994e9ddbp-113},
{0x1.11301d0125b51p0, -0x1.6c51039449b3ap-54, 0x1.9d58b988f562dp-109},
{0x1.1429aaea92dep0, -0x1.32fbf9af1369ep-54, -0x1.2fe7bb4c76416p-108},
{0x1.172b83c7d517bp0, -0x1.19041b9d78a76p-55, 0x1.4f2406aa13ffp-109},
{0x1.1a35beb6fcb75p0, 0x1.e5b4c7b4968e4p-55, 0x1.ad36183926ae8p-111},
{0x1.1d4873168b9aap0, 0x1.e016e00a2643cp-54, 0x1.ea62d0881b918p-110},
{0x1.2063b88628cd6p0, 0x1.dc775814a8495p-55, -0x1.781dbc16f1ea4p-111},
{0x1.2387a6e756238p0, 0x1.9b07eb6c70573p-54, -0x1.4d89f9af532ep-109},
{0x1.26b4565e27cddp0, 0x1.2bd339940e9d9p-55, 0x1.277393a461b77p-110},
{0x1.29e9df51fdee1p0, 0x1.612e8afad1255p-55, 0x1.de5448560469p-111},
{0x1.2d285a6e4030bp0, 0x1.0024754db41d5p-54, -0x1.ee9d8f8cb9307p-110},
{0x1.306fe0a31b715p0, 0x1.6f46ad23182e4p-55, 0x1.7b7b2f09cd0d9p-110},
{0x1.33c08b26416ffp0, 0x1.32721843659a6p-54, -0x1.406a2ea6cfc6bp-108},
{0x1.371a7373aa9cbp0, -0x1.63aeabf42eae2p-54, 0x1.87e3e12516bfap-108},
{0x1.3a7db34e59ff7p0, -0x1.5e436d661f5e3p-56, 0x1.9b0b1ff17c296p-111},
{0x1.3dea64c123422p0, 0x1.ada0911f09ebcp-55, -0x1.808ba68fa8fb7p-109},
{0x1.4160a21f72e2ap0, -0x1.ef3691c309278p-58, -0x1.32b43eafc6518p-114},
{0x1.44e086061892dp0, 0x1.89b7a04ef80dp-59, -0x1.0ac312de3d922p-114},
{0x1.486a2b5c13cdp0, 0x1.3c1a3b69062fp-56, 0x1.e1eebae743acp-111},
{0x1.4bfdad5362a27p0, 0x1.d4397afec42e2p-56, 0x1.c06c7745c2b39p-113},
{0x1.4f9b2769d2ca7p0, -0x1.4b309d25957e3p-54, -0x1.1aa1fd7b685cdp-112},
{0x1.5342b569d4f82p0, -0x1.07abe1db13cadp-55, 0x1.fa733951f214cp-111},
{0x1.56f4736b527dap0, 0x1.9bb2c011d93adp-54, -0x1.ff86852a613ffp-111},
{0x1.5ab07dd485429p0, 0x1.6324c054647adp-54, -0x1.744ee506fdafep-109},
{0x1.5e76f15ad2148p0, 0x1.ba6f93080e65ep-54, -0x1.95f9ab75fa7d6p-108},
{0x1.6247eb03a5585p0, -0x1.383c17e40b497p-54, 0x1.5d8e757cfb991p-111},
{0x1.6623882552225p0, -0x1.bb60987591c34p-54, 0x1.4a337f4dc0a3bp-108},
{0x1.6a09e667f3bcdp0, -0x1.bdd3413b26456p-54, 0x1.57d3e3adec175p-108},
{0x1.6dfb23c651a2fp0, -0x1.bbe3a683c88abp-57, 0x1.a59f88abbe778p-115},
{0x1.71f75e8ec5f74p0, -0x1.16e4786887a99p-55, -0x1.269796953a4c3p-109},
{0x1.75feb564267c9p0, -0x1.0245957316dd3p-54, -0x1.8f8e7fa19e5e8p-108},
{0x1.7a11473eb0187p0, -0x1.41577ee04992fp-55, -0x1.4217a932d10d4p-113},
{0x1.7e2f336cf4e62p0, 0x1.05d02ba15797ep-56, 0x1.70a1427f8fcdfp-112},
{0x1.82589994cce13p0, -0x1.d4c1dd41532d8p-54, 0x1.0f6ad65cbbac1p-112},
{0x1.868d99b4492edp0, -0x1.fc6f89bd4f6bap-54, -0x1.f16f65181d921p-109},
{0x1.8ace5422aa0dbp0, 0x1.6e9f156864b27p-54, -0x1.30644a7836333p-110},
{0x1.8f1ae99157736p0, 0x1.5cc13a2e3976cp-55, 0x1.3bf26d2b85163p-114},
{0x1.93737b0cdc5e5p0, -0x1.75fc781b57ebcp-57, 0x1.697e257ac0db2p-111},
{0x1.97d829fde4e5p0, -0x1.d185b7c1b85d1p-54, 0x1.7edb9d7144b6fp-108},
{0x1.9c49182a3f09p0, 0x1.c7c46b071f2bep-56, 0x1.6376b7943085cp-110},
{0x1.a0c667b5de565p0, -0x1.359495d1cd533p-54, 0x1.354084551b4fbp-109},
{0x1.a5503b23e255dp0, -0x1.d2f6edb8d41e1p-54, -0x1.bfd7adfd63f48p-111},
{0x1.a9e6b5579fdbfp0, 0x1.0fac90ef7fd31p-54, 0x1.8b16ae39e8cb9p-109},
{0x1.ae89f995ad3adp0, 0x1.7a1cd345dcc81p-54, 0x1.a7fbc3ae675eap-108},
{0x1.b33a2b84f15fbp0, -0x1.2805e3084d708p-57, 0x1.2babc0edda4d9p-111},
{0x1.b7f76f2fb5e47p0, -0x1.5584f7e54ac3bp-56, 0x1.aa64481e1ab72p-111},
{0x1.bcc1e904bc1d2p0, 0x1.23dd07a2d9e84p-55, 0x1.9a164050e1258p-109},
{0x1.c199bdd85529cp0, 0x1.11065895048ddp-55, 0x1.99e51125928dap-110},
{0x1.c67f12e57d14bp0, 0x1.2884dff483cadp-54, -0x1.fc44c329d5cb2p-109},
{0x1.cb720dcef9069p0, 0x1.503cbd1e949dbp-56, 0x1.d8765566b032ep-110},
{0x1.d072d4a07897cp0, -0x1.cbc3743797a9cp-54, -0x1.e7044039da0f6p-108},
{0x1.d5818dcfba487p0, 0x1.2ed02d75b3707p-55, -0x1.ab053b05531fcp-111},
{0x1.da9e603db3285p0, 0x1.c2300696db532p-54, 0x1.7f6246f0ec615p-108},
{0x1.dfc97337b9b5fp0, -0x1.1a5cd4f184b5cp-54, 0x1.b7225a944efd6p-108},
{0x1.e502ee78b3ff6p0, 0x1.39e8980a9cc8fp-55, 0x1.1e92cb3c2d278p-109},
{0x1.ea4afa2a490dap0, -0x1.e9c23179c2893p-54, -0x1.fc0f242bbf3dep-109},
{0x1.efa1bee615a27p0, 0x1.dc7f486a4b6bp-54, 0x1.f6dd5d229ff69p-108},
{0x1.f50765b6e454p0, 0x1.9d3e12dd8a18bp-54, -0x1.4019bffc80ef3p-110},
{0x1.fa7c1819e90d8p0, 0x1.74853f3a5931ep-55, 0x1.dc060c36f7651p-112},
};

// 2^(k * 2^-12), for k = 0..63.
constexpr TripleDouble EXP_MID2[64] = {
{0x1p0, 0, 0},
{0x1.000b175effdc7p0, 0x1.ae8e38c59c72ap-54, 0x1.39726694630e3p-108},
{0x1.00162f3904052p0, -0x1.7b5d0d58ea8f4p-58, 0x1.e5e06ddd31156p-112},
{0x1.0021478e11ce6p0, 0x1.4115cb6b16a8ep-54, 0x1.5a0768b51f609p-111},
{0x1.002c605e2e8cfp0, -0x1.d7c96f201bb2fp-55, 0x1.d008403605217p-111},
{0x1.003779a95f959p0, 0x1.84711d4c35e9fp-54, 0x1.89bc16f765708p-109},
{0x1.0042936faa3d8p0, -0x1.0484245243777p-55, -0x1.4535b7f8c1e2dp-109},
{0x1.004dadb113dap0, -0x1.4b237da2025f9p-54, -0x1.8ba92f6b25456p-108},
{0x1.0058c86da1c0ap0, -0x1.5e00e62d6b30dp-56, -0x1.30c72e81f4294p-113},
{0x1.0063e3a559473p0, 0x1.a1d6cedbb9481p-54, -0x1.34a5384e6f0b9p-110},
{0x1.006eff583fc3dp0, -0x1.4acf197a00142p-54, 0x1.f8d0580865d2ep-108},
{0x1.007a1b865a8cap0, -0x1.eaf2ea42391a5p-57, -0x1.002bcb3ae9a99p-111},
{0x1.0085382faef83p0, 0x1.da93f90835f75p-56, 0x1.c3c5aedee9851p-111},
{0x1.00905554425d4p0, -0x1.6a79084ab093cp-55, 0x1.7217851d1ec6ep-109},
{0x1.009b72f41a12bp0, 0x1.86364f8fbe8f8p-54, -0x1.80cbca335a7c3p-110},
{0x1.00a6910f3b6fdp0, -0x1.82e8e14e3110ep-55, -0x1.706bd4eb22595p-110},
{0x1.00b1afa5abcbfp0, -0x1.4f6b2a7609f71p-55, -0x1.b55dd523f3c08p-111},
{0x1.00bcceb7707ecp0, -0x1.e1a258ea8f71bp-56, 0x1.90a1e207cced1p-110},
{0x1.00c7ee448ee02p0, 0x1.4362ca5bc26f1p-56, 0x1.78d0472db37c5p-110},
{0x1.00d30e4d0c483p0, 0x1.095a56c919d02p-54, -0x1.bcd4db3cb52fep-109},
{0x1.00de2ed0ee0f5p0, -0x1.406ac4e81a645p-57, -0x1.cf1b131575ec2p-112},
{0x1.00e94fd0398ep0, 0x1.b5a6902767e09p-54, -0x1.6aaa1fa7ff913p-112},
{0x1.00f4714af41d3p0, -0x1.91b2060859321p-54, 0x1.68f236dff3218p-110},
{0x1.00ff93412315cp0, 0x1.427068ab22306p-55, -0x1.e8bb58067e60ap-109},
{0x1.010ab5b2cbd11p0, 0x1.c1d0660524e08p-54, 0x1.d4cd5e1d71fdfp-108},
{0x1.0115d89ff3a8bp0, -0x1.e7bdfb3204be8p-54, 0x1.e4ecf350ebe88p-108},
{0x1.0120fc089ff63p0, 0x1.843aa8b9cbbc6p-55, 0x1.6a2aa2c89c4f8p-109},
{0x1.012c1fecd613bp0, -0x1.34104ee7edae9p-56, 0x1.1ca368a20ed05p-110},
{0x1.0137444c9b5b5p0, -0x1.2b6aeb6176892p-56, 0x1.edb1095d925cfp-114},
{0x1.01426927f5278p0, 0x1.a8cd33b8a1bb3p-56, -0x1.488c78eded75fp-111},
{0x1.014d8e7ee8d2fp0, 0x1.2edc08e5da99ap-56, -0x1.7480f5ea1b3c9p-113},
{0x1.0158b4517bb88p0, 0x1.57ba2dc7e0c73p-55, -0x1.ae45989a04dd5p-111},
{0x1.0163da9fb3335p0, 0x1.b61299ab8cdb7p-54, 0x1.bf48007d80987p-109},
{0x1.016f0169949edp0, -0x1.90565902c5f44p-54, 0x1.1aa91a059292cp-109},
{0x1.017a28af25567p0, 0x1.70fc41c5c2d53p-55, 0x1.b6663292855f5p-110},
{0x1.018550706ab62p0, 0x1.4b9a6e145d76cp-54, 0x1.e7fbca6793d94p-108},
{0x1.019078ad6a19fp0, -0x1.008eff5142bf9p-56, -0x1.5b9f5c7de3b93p-110},
{0x1.019ba16628de2p0, -0x1.77669f033c7dep-54, 0x1.4638bf2f6acabp-110},
{0x1.01a6ca9aac5f3p0, -0x1.09bb78eeead0ap-54, -0x1.ab237b9a069c5p-109},
{0x1.01b1f44af9f9ep0, 0x1.371231477ece5p-54, 0x1.3ab358be97cefp-108},
{0x1.01bd1e77170b4p0, 0x1.5e7626621eb5bp-56, -0x1.4027b2294bb64p-110},
{0x1.01c8491f08f08p0, -0x1.bc72b100828a5p-54, 0x1.656394426c99p-111},
{0x1.01d37442d507p0, -0x1.ce39cbbab8bbep-57, 0x1.bf9785189bdd8p-111},
{0x1.01de9fe280ac8p0, 0x1.16996709da2e2p-55, 0x1.7c12f86114fe3p-109},
{0x1.01e9cbfe113efp0, -0x1.c11f5239bf535p-55, -0x1.653d5d24b5d28p-109},
{0x1.01f4f8958c1c6p0, 0x1.e1d4eb5edc6b3p-55, 0x1.04a0cdc1d86d7p-109},
{0x1.020025a8f6a35p0, -0x1.afb99946ee3fp-54, 0x1.c678c46149782p-109},
{0x1.020b533856324p0, -0x1.8f06d8a148a32p-54, 0x1.48524e1e9df7p-108},
{0x1.02168143b0281p0, -0x1.2bf310fc54eb6p-55, 0x1.9953ea727ff0bp-109},
{0x1.0221afcb09e3ep0, -0x1.c95a035eb4175p-54, -0x1.ccfbbec22d28ep-108},
{0x1.022cdece68c4fp0, -0x1.491793e46834dp-54, 0x1.9e2bb6e181de1p-108},
{0x1.02380e4dd22adp0, -0x1.3e8d0d9c49091p-56, 0x1.f17609ae29308p-110},
{0x1.02433e494b755p0, -0x1.314aa16278aa3p-54, -0x1.c7dc2c476bfb8p-110},
{0x1.024e6ec0da046p0, 0x1.48daf888e9651p-55, -0x1.fab994971d4a3p-109},
{0x1.02599fb483385p0, 0x1.56dc8046821f4p-55, 0x1.848b62cbdd0afp-109},
{0x1.0264d1244c719p0, 0x1.45b42356b9d47p-54, -0x1.bf603ba715d0cp-109},
{0x1.027003103b10ep0, -0x1.082ef51b61d7ep-56, 0x1.89434e751e1aap-110},
{0x1.027b357854772p0, 0x1.2106ed0920a34p-56, -0x1.03b54fd64e8acp-110},
{0x1.0286685c9e059p0, -0x1.fd4cf26ea5d0fp-54, 0x1.7785ea0acc486p-109},
{0x1.02919bbd1d1d8p0, -0x1.09f8775e78084p-54, -0x1.ce447fdb35ff9p-109},
{0x1.029ccf99d720ap0, 0x1.64cbba902ca27p-58, 0x1.5b884aab5642ap-112},
{0x1.02a803f2d170dp0, 0x1.4383ef231d207p-54, -0x1.cfb3e46d7c1cp-108},
{0x1.02b338c811703p0, 0x1.4a47a505b3a47p-54, -0x1.0d40cee4b81afp-112},
{0x1.02be6e199c811p0, 0x1.e47120223467fp-54, 0x1.6ae7d36d7c1f7p-109},
};

// Polynomial approximations with double precision:
// Return expm1(dx) / x ~ 1 + dx / 2 + dx^2 / 6 + dx^3 / 24.
// For |dx| < 2^-13 + 2^-30:
Expand Down Expand Up @@ -267,14 +128,14 @@ Float128 exp_f128(double x, double kd, int idx1, int idx2) {

// TODO: Skip recalculating exp_mid1 and exp_mid2.
Float128 exp_mid1 =
fputil::quick_add(Float128(EXP_MID1[idx1].hi),
fputil::quick_add(Float128(EXP_MID1[idx1].mid),
Float128(EXP_MID1[idx1].lo)));
fputil::quick_add(Float128(EXP2_MID1[idx1].hi),
fputil::quick_add(Float128(EXP2_MID1[idx1].mid),
Float128(EXP2_MID1[idx1].lo)));

Float128 exp_mid2 =
fputil::quick_add(Float128(EXP_MID2[idx2].hi),
fputil::quick_add(Float128(EXP_MID2[idx2].mid),
Float128(EXP_MID2[idx2].lo)));
fputil::quick_add(Float128(EXP2_MID2[idx2].hi),
fputil::quick_add(Float128(EXP2_MID2[idx2].mid),
Float128(EXP2_MID2[idx2].lo)));

Float128 exp_mid = fputil::quick_mul(exp_mid1, exp_mid2);

Expand Down Expand Up @@ -309,48 +170,8 @@ DoubleDouble exp_double_double(double x, double kd,
return r;
}

// Rounding tests when the output might be denormal.
cpp::optional<double> ziv_test_denorm(int hi, double mid, double lo,
double err) {
using FloatProp = typename fputil::FloatProperties<double>;

// Scaling factor = 1/(min normal number) = 2^1022
int64_t exp_hi = static_cast<int64_t>(hi + 1022) << FloatProp::MANTISSA_WIDTH;
double mid_hi = cpp::bit_cast<double>(exp_hi + cpp::bit_cast<int64_t>(mid));

// Extra errors from another rounding step.
err += 0x1.0p-52;

double lo_u = lo + err;
double lo_l = lo - err;
double mid_lo_u =
cpp::bit_cast<double>(exp_hi + cpp::bit_cast<int64_t>(lo_u));
double mid_lo_l =
cpp::bit_cast<double>(exp_hi + cpp::bit_cast<int64_t>(lo_l));

// By adding 2^-511, the results will have similar rounding points as denormal
// outputs.
double upper = (mid_hi + mid_lo_u);
double lower = (mid_hi + mid_lo_l);

uint64_t scale_down = 0;

if (upper < 1.0) {
// Upper bound is in denormal range, need extra rounding.
upper += 1.0;
lower += 1.0;
scale_down = 0x3FF0'0000'0000'0000; // 1.0
}

if (LIBC_LIKELY(upper == lower)) {
return cpp::bit_cast<double>(cpp::bit_cast<uint64_t>(upper) - scale_down);
}

return cpp::nullopt;
}

// Check for exceptional cases when
// |x| < 2^-53
// |x| <= 2^-53 or x < log(2^-1075) or x >= 0x1.6232bdd7abcd3p+9
double set_exceptional(double x) {
using FPBits = typename fputil::FPBits<double>;
using FloatProp = typename fputil::FloatProperties<double>;
Expand All @@ -359,7 +180,7 @@ double set_exceptional(double x) {
uint64_t x_u = xbits.uintval();
uint64_t x_abs = x_u & FloatProp::EXP_MANT_MASK;

// |x| < 2^-53
// |x| <= 2^-53
if (x_abs <= 0x3ca0'0000'0000'0000ULL) {
// exp(x) ~ 1 + x
return 1 + x;
Expand Down Expand Up @@ -424,7 +245,7 @@ LLVM_LIBC_FUNCTION(double, exp, (double x)) {
return set_exceptional(x);
}

// Now log(2^-1022) <= x <= -2^-53 or 2^-53 <= x < log(2^1023 * (2 - 2^-52))
// Now log(2^-1075) <= x <= -2^-53 or 2^-53 <= x < log(2^1023 * (2 - 2^-52))

// Range reduction:
// Let x = log(2) * (hi + mid1 + mid2) + lo
Expand Down Expand Up @@ -514,8 +335,8 @@ LLVM_LIBC_FUNCTION(double, exp, (double x)) {

bool denorm = (hi <= -1022);

DoubleDouble exp_mid1{EXP_MID1[idx1].mid, EXP_MID1[idx1].hi};
DoubleDouble exp_mid2{EXP_MID2[idx2].mid, EXP_MID2[idx2].hi};
DoubleDouble exp_mid1{EXP2_MID1[idx1].mid, EXP2_MID1[idx1].hi};
DoubleDouble exp_mid2{EXP2_MID2[idx2].mid, EXP2_MID2[idx2].hi};

DoubleDouble exp_mid = fputil::quick_mult(exp_mid1, exp_mid2);

Expand Down
390 changes: 390 additions & 0 deletions libc/src/math/generic/exp2.cpp
View file
Original file line number Diff line number Diff line change
@@ -0,0 +1,390 @@
//===-- Double-precision 2^x function -------------------------------------===//
//
// Part of the LLVM Project, under the Apache License v2.0 with LLVM Exceptions.
// See https://llvm.org/LICENSE.txt for license information.
// SPDX-License-Identifier: Apache-2.0 WITH LLVM-exception
//
//===----------------------------------------------------------------------===//

#include "src/math/exp2.h"
#include "common_constants.h" // Lookup tables EXP2_MID1 and EXP_M2.
#include "explogxf.h" // ziv_test_denorm.
#include "src/__support/CPP/bit.h"
#include "src/__support/CPP/optional.h"
#include "src/__support/FPUtil/FEnvImpl.h"
#include "src/__support/FPUtil/FPBits.h"
#include "src/__support/FPUtil/PolyEval.h"
#include "src/__support/FPUtil/double_double.h"
#include "src/__support/FPUtil/dyadic_float.h"
#include "src/__support/FPUtil/multiply_add.h"
#include "src/__support/FPUtil/nearest_integer.h"
#include "src/__support/FPUtil/rounding_mode.h"
#include "src/__support/FPUtil/triple_double.h"
#include "src/__support/common.h"
#include "src/__support/macros/optimization.h" // LIBC_UNLIKELY

#include <errno.h>

namespace __llvm_libc {

using fputil::DoubleDouble;
using fputil::TripleDouble;
using Float128 = typename fputil::DyadicFloat<128>;

// Error bounds:
// Errors when using double precision.
#ifdef LIBC_TARGET_CPU_HAS_FMA
constexpr double ERR_D = 0x1.0p-63;
#else
constexpr double ERR_D = 0x1.8p-63;
#endif // LIBC_TARGET_CPU_HAS_FMA

// Errors when using double-double precision.
constexpr double ERR_DD = 0x1.0p-100;

// Polynomial approximations with double precision. Generated by Sollya with:
// > P = fpminimax((2^x - 1)/x, 3, [|D...|], [-2^-13 - 2^-30, 2^-13 + 2^-30]);
// > P;
// Error bounds:
// | output - (2^dx - 1) / dx | < 1.5 * 2^-52.
LIBC_INLINE double poly_approx_d(double dx) {
// dx^2
double dx2 = dx * dx;
double c0 =
fputil::multiply_add(dx, 0x1.ebfbdff82c58ep-3, 0x1.62e42fefa39efp-1);
double c1 =
fputil::multiply_add(dx, 0x1.3b2aba7a95a89p-7, 0x1.c6b08e8fc0c0ep-5);
double p = fputil::multiply_add(dx2, c1, c0);
return p;
}

// Polynomial approximation with double-double precision. Generated by Solya
// with:
// > P = fpminimax((2^x - 1)/x, 5, [|DD...|], [-2^-13 - 2^-30, 2^-13 + 2^-30]);
// Error bounds:
// | output - 2^(dx) | < 2^-101
DoubleDouble poly_approx_dd(const DoubleDouble &dx) {
// Taylor polynomial.
constexpr DoubleDouble COEFFS[] = {
{0, 0x1p0},
{0x1.abc9e3b39824p-56, 0x1.62e42fefa39efp-1},
{-0x1.5e43a53e4527bp-57, 0x1.ebfbdff82c58fp-3},
{-0x1.d37963a9444eep-59, 0x1.c6b08d704a0cp-5},
{0x1.4eda1a81133dap-62, 0x1.3b2ab6fba4e77p-7},
{-0x1.c53fd1ba85d14p-64, 0x1.5d87fe7a265a5p-10},
{0x1.d89250b013eb8p-70, 0x1.430912f86cb8ep-13},
};

DoubleDouble p = fputil::polyeval(dx, COEFFS[0], COEFFS[1], COEFFS[2],
COEFFS[3], COEFFS[4], COEFFS[5], COEFFS[6]);
return p;
}

// Polynomial approximation with 128-bit precision:
// Return exp(dx) ~ 1 + a0 * dx + a1 * dx^2 + ... + a6 * dx^7
// For |dx| < 2^-13 + 2^-30:
// | output - exp(dx) | < 2^-126.
Float128 poly_approx_f128(const Float128 &dx) {
using MType = typename Float128::MantissaType;

constexpr Float128 COEFFS_128[]{
{false, -127, MType({0, 0x8000000000000000})}, // 1.0
{false, -128, MType({0xc9e3b39803f2f6af, 0xb17217f7d1cf79ab})},
{false, -128, MType({0xde2d60dd9c9a1d9f, 0x3d7f7bff058b1d50})},
{false, -132, MType({0x9d3b15d9e7fb6897, 0xe35846b82505fc59})},
{false, -134, MType({0x184462f6bcd2b9e7, 0x9d955b7dd273b94e})},
{false, -137, MType({0x39ea1bb964c51a89, 0xaec3ff3c53398883})},
{false, -138, MType({0x842c53418fa8ae61, 0x2861225f345c396a})},
{false, -144, MType({0x7abeb5abd5ad2079, 0xffe5fe2d109a319d})},
};

Float128 p = fputil::polyeval(dx, COEFFS_128[0], COEFFS_128[1], COEFFS_128[2],
COEFFS_128[3], COEFFS_128[4], COEFFS_128[5],
COEFFS_128[6], COEFFS_128[7]);
return p;
}

// Compute exp(x) using 128-bit precision.
// TODO(lntue): investigate triple-double precision implementation for this
// step.
Float128 exp2_f128(double x, int hi, int idx1, int idx2) {
Float128 dx = Float128(x);

// TODO: Skip recalculating exp_mid1 and exp_mid2.
Float128 exp_mid1 =
fputil::quick_add(Float128(EXP2_MID1[idx1].hi),
fputil::quick_add(Float128(EXP2_MID1[idx1].mid),
Float128(EXP2_MID1[idx1].lo)));

Float128 exp_mid2 =
fputil::quick_add(Float128(EXP2_MID2[idx2].hi),
fputil::quick_add(Float128(EXP2_MID2[idx2].mid),
Float128(EXP2_MID2[idx2].lo)));

Float128 exp_mid = fputil::quick_mul(exp_mid1, exp_mid2);

Float128 p = poly_approx_f128(dx);

Float128 r = fputil::quick_mul(exp_mid, p);

r.exponent += hi;

return r;
}

// Compute 2^x with double-double precision.
DoubleDouble exp2_double_double(double x, const DoubleDouble &exp_mid) {
DoubleDouble dx({0, x});

// Degree-6 polynomial approximation in double-double precision.
// | p - 2^x | < 2^-103.
DoubleDouble p = poly_approx_dd(dx);

// Error bounds: 2^-102.
DoubleDouble r = fputil::quick_mult(exp_mid, p);

return r;
}

// When output is denormal.
double exp2_denorm(double x) {
// Range reduction.
int k =
static_cast<int>(cpp::bit_cast<uint64_t>(x + 0x1.8000'0000'4p21) >> 19);
double kd = static_cast<double>(k);

uint32_t idx1 = (k >> 6) & 0x3f;
uint32_t idx2 = k & 0x3f;

int hi = k >> 12;

DoubleDouble exp_mid1{EXP2_MID1[idx1].mid, EXP2_MID1[idx1].hi};
DoubleDouble exp_mid2{EXP2_MID2[idx2].mid, EXP2_MID2[idx2].hi};
DoubleDouble exp_mid = fputil::quick_mult(exp_mid1, exp_mid2);

// |dx| < 2^-13 + 2^-30.
double dx = fputil::multiply_add(kd, -0x1.0p-12, x); // exact

double mid_lo = dx * exp_mid.hi;

// Approximate (2^dx - 1)/dx ~ 1 + a0*dx + a1*dx^2 + a2*dx^3 + a3*dx^4.
double p = poly_approx_d(dx);

double lo = fputil::multiply_add(p, mid_lo, exp_mid.lo);

if (auto r = ziv_test_denorm(hi, exp_mid.hi, lo, ERR_D);
LIBC_LIKELY(r.has_value()))
return r.value();

// Use double-double
DoubleDouble r_dd = exp2_double_double(dx, exp_mid);

if (auto r = ziv_test_denorm(hi, r_dd.hi, r_dd.lo, ERR_DD);
LIBC_LIKELY(r.has_value()))
return r.value();

// Use 128-bit precision
Float128 r_f128 = exp2_f128(dx, hi, idx1, idx2);

return static_cast<double>(r_f128);
}

// Check for exceptional cases when:
// * log2(1 - 2^-54) < x < log2(1 + 2^-53)
// * x >= 1024
// * x <= -1075
// * x is inf or nan
double set_exceptional(double x) {
using FPBits = typename fputil::FPBits<double>;
using FloatProp = typename fputil::FloatProperties<double>;
FPBits xbits(x);

uint64_t x_u = xbits.uintval();
uint64_t x_abs = x_u & FloatProp::EXP_MANT_MASK;

// |x| < log2(1 + 2^-53)
if (x_abs <= 0x3ca71547652b82fd) {
// 2^(x) ~ 1 + x/2
return fputil::multiply_add(x, 0.5, 1.0);
}

// x <= 2^-1075 || x >= 1024 or inf/nan.
if (x_u > 0xc08ff00000000000) {
// x <= 2^-1075 or -inf/nan
if (x_u >= 0xc090cc0000000000) {
// exp(-Inf) = 0
if (xbits.is_inf())
return 0.0;

// exp(nan) = nan
if (xbits.is_nan())
return x;

if (fputil::quick_get_round() == FE_UPWARD)
return static_cast<double>(FPBits(FPBits::MIN_SUBNORMAL));
fputil::set_errno_if_required(ERANGE);
fputil::raise_except_if_required(FE_UNDERFLOW);
return 0.0;
}

return exp2_denorm(x);
}

// x >= 1024 or +inf/nan
// x is finite
if (x_u < 0x7ff0'0000'0000'0000ULL) {
int rounding = fputil::quick_get_round();
if (rounding == FE_DOWNWARD || rounding == FE_TOWARDZERO)
return static_cast<double>(FPBits(FPBits::MAX_NORMAL));

fputil::set_errno_if_required(ERANGE);
fputil::raise_except_if_required(FE_OVERFLOW);
}
// x is +inf or nan
return x + static_cast<double>(FPBits::inf());
}

LLVM_LIBC_FUNCTION(double, exp2, (double x)) {
using FPBits = typename fputil::FPBits<double>;
using FloatProp = typename fputil::FloatProperties<double>;
FPBits xbits(x);

uint64_t x_u = xbits.uintval();

// x < -1022 or x >= 1024 or log2(1 - 2^-54) < x < log2(1 + 2^-53).
if (LIBC_UNLIKELY(x_u > 0xc08ff00000000000 ||
(x_u <= 0xbc971547652b82fe && x_u >= 0x4090000000000000) ||
x_u <= 0x3ca71547652b82fd)) {
return set_exceptional(x);
}

// Now -1075 < x <= log2(1 - 2^-54) or log2(1 + 2^-53) < x < 1024

// Range reduction:
// Let x = (hi + mid1 + mid2) + lo
// in which:
// hi is an integer
// mid1 * 2^6 is an integer
// mid2 * 2^12 is an integer
// then:
// 2^(x) = 2^hi * 2^(mid1) * 2^(mid2) * 2^(lo).
// With this formula:
// - multiplying by 2^hi is exact and cheap, simply by adding the exponent
// field.
// - 2^(mid1) and 2^(mid2) are stored in 2 x 64-element tables.
// - 2^(lo) ~ 1 + a0*lo + a1 * lo^2 + ...
//
// We compute (hi + mid1 + mid2) together by perform the rounding on x * 2^12.
// Since |x| < |-1075)| < 2^11,
// |x * 2^12| < 2^11 * 2^12 < 2^23,
// So we can fit the rounded result round(x * 2^12) in int32_t.
// Thus, the goal is to be able to use an additional addition and fixed width
// shift to get an int32_t representing round(x * 2^12).
//
// Assuming int32_t using 2-complement representation, since the mantissa part
// of a double precision is unsigned with the leading bit hidden, if we add an
// extra constant C = 2^e1 + 2^e2 with e1 > e2 >= 2^25 to the product, the
// part that are < 2^e2 in resulted mantissa of (x*2^12*L2E + C) can be
// considered as a proper 2-complement representations of x*2^12.
//
// One small problem with this approach is that the sum (x*2^12 + C) in
// double precision is rounded to the least significant bit of the dorminant
// factor C. In order to minimize the rounding errors from this addition, we
// want to minimize e1. Another constraint that we want is that after
// shifting the mantissa so that the least significant bit of int32_t
// corresponds to the unit bit of (x*2^12*L2E), the sign is correct without
// any adjustment. So combining these 2 requirements, we can choose
// C = 2^33 + 2^32, so that the sign bit corresponds to 2^31 bit, and hence
// after right shifting the mantissa, the resulting int32_t has correct sign.
// With this choice of C, the number of mantissa bits we need to shift to the
// right is: 52 - 33 = 19.
//
// Moreover, since the integer right shifts are equivalent to rounding down,
// we can add an extra 0.5 so that it will become round-to-nearest, tie-to-
// +infinity. So in particular, we can compute:
// hmm = x * 2^12 + C,
// where C = 2^33 + 2^32 + 2^-1, then if
// k = int32_t(lower 51 bits of double(x * 2^12 + C) >> 19),
// the reduced argument:
// lo = x - 2^-12 * k is bounded by:
// |lo| <= 2^-13 + 2^-12*2^-19
// = 2^-13 + 2^-31.
//
// Finally, notice that k only uses the mantissa of x * 2^12, so the
// exponent 2^12 is not needed. So we can simply define
// C = 2^(33 - 12) + 2^(32 - 12) + 2^(-13 - 12), and
// k = int32_t(lower 51 bits of double(x + C) >> 19).

// Rounding errors <= 2^-31.
int k =
static_cast<int>(cpp::bit_cast<uint64_t>(x + 0x1.8000'0000'4p21) >> 19);
double kd = static_cast<double>(k);

uint32_t idx1 = (k >> 6) & 0x3f;
uint32_t idx2 = k & 0x3f;

int hi = k >> 12;

DoubleDouble exp_mid1{EXP2_MID1[idx1].mid, EXP2_MID1[idx1].hi};
DoubleDouble exp_mid2{EXP2_MID2[idx2].mid, EXP2_MID2[idx2].hi};
DoubleDouble exp_mid = fputil::quick_mult(exp_mid1, exp_mid2);

// |dx| < 2^-13 + 2^-30.
double dx = fputil::multiply_add(kd, -0x1.0p-12, x); // exact

// We use the degree-4 polynomial to approximate 2^(lo):
// 2^(lo) ~ 1 + a0 * lo + a1 * lo^2 + a2 * lo^3 + a3 * lo^4 = 1 + lo * P(lo)
// So that the errors are bounded by:
// |P(lo) - (2^lo - 1)/lo| < |lo|^4 / 64 < 2^(-13 * 4) / 64 = 2^-58
// Let P_ be an evaluation of P where all intermediate computations are in
// double precision. Using either Horner's or Estrin's schemes, the evaluated
// errors can be bounded by:
// |P_(lo) - P(lo)| < 2^-51
// => |lo * P_(lo) - (2^lo - 1) | < 2^-64
// => 2^(mid1 + mid2) * |lo * P_(lo) - expm1(lo)| < 2^-63.
// Since we approximate
// 2^(mid1 + mid2) ~ exp_mid.hi + exp_mid.lo,
// We use the expression:
// (exp_mid.hi + exp_mid.lo) * (1 + dx * P_(dx)) ~
// ~ exp_mid.hi + (exp_mid.hi * dx * P_(dx) + exp_mid.lo)
// with errors bounded by 2^-63.

double mid_lo = dx * exp_mid.hi;

// Approximate (2^dx - 1)/dx ~ 1 + a0*dx + a1*dx^2 + a2*dx^3 + a3*dx^4.
double p = poly_approx_d(dx);

double lo = fputil::multiply_add(p, mid_lo, exp_mid.lo);

double upper = exp_mid.hi + (lo + ERR_D);
double lower = exp_mid.hi + (lo - ERR_D);

if (LIBC_LIKELY(upper == lower)) {
// To multiply by 2^hi, a fast way is to simply add hi to the exponent
// field.
int64_t exp_hi = static_cast<int64_t>(hi) << FloatProp::MANTISSA_WIDTH;
double r = cpp::bit_cast<double>(exp_hi + cpp::bit_cast<int64_t>(upper));
return r;
}

// Use double-double
DoubleDouble r_dd = exp2_double_double(dx, exp_mid);

double upper_dd = r_dd.hi + (r_dd.lo + ERR_DD);
double lower_dd = r_dd.hi + (r_dd.lo - ERR_DD);

if (LIBC_LIKELY(upper_dd == lower_dd)) {
// To multiply by 2^hi, a fast way is to simply add hi to the exponent
// field.
int64_t exp_hi = static_cast<int64_t>(hi) << FloatProp::MANTISSA_WIDTH;
double r = cpp::bit_cast<double>(exp_hi + cpp::bit_cast<int64_t>(upper_dd));
return r;
}

// Use 128-bit precision
Float128 r_f128 = exp2_f128(dx, hi, idx1, idx2);

return static_cast<double>(r_f128);
}

} // namespace __llvm_libc
48 changes: 48 additions & 0 deletions libc/src/math/generic/explogxf.h
View file
Original file line number Diff line number Diff line change
Expand Up @@ -11,6 +11,8 @@

#include "common_constants.h"
#include "math_utils.h"
#include "src/__support/CPP/bit.h"
#include "src/__support/CPP/optional.h"
#include "src/__support/FPUtil/FEnvImpl.h"
#include "src/__support/FPUtil/FPBits.h"
#include "src/__support/FPUtil/PolyEval.h"
Expand Down Expand Up @@ -333,6 +335,52 @@ LIBC_INLINE static double log_eval(double x) {
return result;
}

// Rounding tests for 2^hi * (mid + lo) when the output might be denormal. We
// assume further that 1 <= mid < 2, mid + lo < 2, and |lo| << mid.
// Notice that, if 0 < x < 2^-1022,
// double(2^-1022 + x) - 2^-1022 = double(x).
// So if we scale x up by 2^1022, we can use
// double(1.0 + 2^1022 * x) - 1.0 to test how x is rounded in denormal range.
LIBC_INLINE cpp::optional<double> ziv_test_denorm(int hi, double mid, double lo,
double err) {
using FloatProp = typename fputil::FloatProperties<double>;

// Scaling factor = 1/(min normal number) = 2^1022
int64_t exp_hi = static_cast<int64_t>(hi + 1022) << FloatProp::MANTISSA_WIDTH;
double mid_hi = cpp::bit_cast<double>(exp_hi + cpp::bit_cast<int64_t>(mid));
double lo_scaled =
(lo != 0.0) ? cpp::bit_cast<double>(exp_hi + cpp::bit_cast<int64_t>(lo))
: 0.0;

double extra_factor = 0.0;
uint64_t scale_down = 0x3FE0'0000'0000'0000; // 1022 in the exponent field.

// Result is denormal if (mid_hi + lo_scale < 1.0).
if ((1.0 - mid_hi) > lo_scaled) {
// Extra rounding step is needed, which adds more rounding errors.
err += 0x1.0p-52;
extra_factor = 1.0;
scale_down = 0x3FF0'0000'0000'0000; // 1023 in the exponent field.
}

double err_scaled =
cpp::bit_cast<double>(exp_hi + cpp::bit_cast<int64_t>(err));

double lo_u = lo_scaled + err_scaled;
double lo_l = lo_scaled - err_scaled;

// By adding 1.0, the results will have similar rounding points as denormal
// outputs.
double upper = extra_factor + (mid_hi + lo_u);
double lower = extra_factor + (mid_hi + lo_l);

if (LIBC_LIKELY(upper == lower)) {
return cpp::bit_cast<double>(cpp::bit_cast<uint64_t>(upper) - scale_down);
}

return cpp::nullopt;
}

} // namespace __llvm_libc

#endif // LLVM_LIBC_SRC_MATH_GENERIC_EXPLOGXF_H
14 changes: 14 additions & 0 deletions libc/test/src/math/CMakeLists.txt
View file
Original file line number Diff line number Diff line change
Expand Up @@ -619,6 +619,20 @@ add_fp_unittest(
libc.src.__support.FPUtil.fp_bits
)

add_fp_unittest(
exp2_test
NEED_MPFR
SUITE
libc_math_unittests
SRCS
exp2_test.cpp
DEPENDS
libc.src.errno.errno
libc.include.math
libc.src.math.exp2
libc.src.__support.FPUtil.fp_bits
)

add_fp_unittest(
exp10f_test
NEED_MPFR
Expand Down
125 changes: 125 additions & 0 deletions libc/test/src/math/exp2_test.cpp
View file
Original file line number Diff line number Diff line change
@@ -0,0 +1,125 @@
//===-- Unittests for 2^x -------------------------------------------------===//
//
// Part of the LLVM Project, under the Apache License v2.0 with LLVM Exceptions.
// See https://llvm.org/LICENSE.txt for license information.
// SPDX-License-Identifier: Apache-2.0 WITH LLVM-exception
//
//===----------------------------------------------------------------------===//

#include "src/__support/FPUtil/FPBits.h"
#include "src/errno/libc_errno.h"
#include "src/math/exp2.h"
#include "test/UnitTest/FPMatcher.h"
#include "test/UnitTest/Test.h"
#include "utils/MPFRWrapper/MPFRUtils.h"
#include <math.h>

#include <errno.h>
#include <stdint.h>

namespace mpfr = __llvm_libc::testing::mpfr;
using __llvm_libc::testing::tlog;

DECLARE_SPECIAL_CONSTANTS(double)

TEST(LlvmLibcExp2Test, SpecialNumbers) {
EXPECT_FP_EQ(aNaN, __llvm_libc::exp2(aNaN));
EXPECT_FP_EQ(inf, __llvm_libc::exp2(inf));
EXPECT_FP_EQ_ALL_ROUNDING(zero, __llvm_libc::exp2(neg_inf));
EXPECT_FP_EQ_WITH_EXCEPTION(zero, __llvm_libc::exp2(-0x1.0p20), FE_UNDERFLOW);
EXPECT_FP_EQ_WITH_EXCEPTION(inf, __llvm_libc::exp2(0x1.0p20), FE_OVERFLOW);
EXPECT_FP_EQ_ALL_ROUNDING(1.0, __llvm_libc::exp2(0.0));
EXPECT_FP_EQ_ALL_ROUNDING(1.0, __llvm_libc::exp2(-0.0));
}

TEST(LlvmLibcExp2Test, TrickyInputs) {
constexpr int N = 16;
constexpr uint64_t INPUTS[N] = {
0x3FD79289C6E6A5C0,
0x3FD05DE80A173EA0, // 0x1.05de80a173eap-2
0xbf1eb7a4cb841fcc, // -0x1.eb7a4cb841fccp-14
0xbf19a61fb925970d,
0x3fda7b764e2cf47a, // 0x1.a7b764e2cf47ap-2
0xc04757852a4b93aa, // -0x1.757852a4b93aap+5
0x4044c19e5712e377, // x=0x1.4c19e5712e377p+5
0xbf19a61fb925970d, // x=-0x1.9a61fb925970dp-14
0xc039a74cdab36c28, // x=-0x1.9a74cdab36c28p+4
0xc085b3e4e2e3bba9, // x=-0x1.5b3e4e2e3bba9p+9
0xc086960d591aec34, // x=-0x1.6960d591aec34p+9
0xc086232c09d58d91, // x=-0x1.6232c09d58d91p+9
0xc0874910d52d3051, // x=-0x1.74910d52d3051p9
0xc0867a172ceb0990, // x=-0x1.67a172ceb099p+9
0xc08ff80000000000, // x=-0x1.ff8p+9
0xbc971547652b82fe, // x=-0x1.71547652b82fep-54
};
for (int i = 0; i < N; ++i) {
double x = double(FPBits(INPUTS[i]));
EXPECT_MPFR_MATCH_ALL_ROUNDING(mpfr::Operation::Exp2, x,
__llvm_libc::exp2(x), 0.5);
}
}

TEST(LlvmLibcExp2Test, InDoubleRange) {
constexpr uint64_t COUNT = 1'231;
uint64_t START = __llvm_libc::fputil::FPBits<double>(0.25).uintval();
uint64_t STOP = __llvm_libc::fputil::FPBits<double>(4.0).uintval();
uint64_t STEP = (STOP - START) / COUNT;

auto test = [&](mpfr::RoundingMode rounding_mode) {
mpfr::ForceRoundingMode __r(rounding_mode);
if (!__r.success)
return;

uint64_t fails = 0;
uint64_t count = 0;
uint64_t cc = 0;
double mx, mr = 0.0;
double tol = 0.5;

for (uint64_t i = 0, v = START; i <= COUNT; ++i, v += STEP) {
double x = FPBits(v).get_val();
if (isnan(x) || isinf(x) || x < 0.0)
continue;
libc_errno = 0;
double result = __llvm_libc::exp2(x);
++cc;
if (isnan(result) || isinf(result))
continue;

++count;

if (!TEST_MPFR_MATCH_ROUNDING_SILENTLY(mpfr::Operation::Exp2, x, result,
0.5, rounding_mode)) {
++fails;
while (!TEST_MPFR_MATCH_ROUNDING_SILENTLY(mpfr::Operation::Exp2, x,
result, tol, rounding_mode)) {
mx = x;
mr = result;

if (tol > 1000.0)
break;

tol *= 2.0;
}
}
}
tlog << " Exp2 failed: " << fails << "/" << count << "/" << cc
<< " tests.\n";
tlog << " Max ULPs is at most: " << static_cast<uint64_t>(tol) << ".\n";
if (fails) {
EXPECT_MPFR_MATCH(mpfr::Operation::Exp2, mx, mr, 0.5, rounding_mode);
}
};

tlog << " Test Rounding To Nearest...\n";
test(mpfr::RoundingMode::Nearest);

tlog << " Test Rounding Downward...\n";
test(mpfr::RoundingMode::Downward);

tlog << " Test Rounding Upward...\n";
test(mpfr::RoundingMode::Upward);

tlog << " Test Rounding Toward Zero...\n";
test(mpfr::RoundingMode::TowardZero);
}