diff --git a/libc/src/math/generic/exp2f.cpp b/libc/src/math/generic/exp2f.cpp --- a/libc/src/math/generic/exp2f.cpp +++ b/libc/src/math/generic/exp2f.cpp @@ -18,54 +18,33 @@ namespace __llvm_libc { -// Lookup table for 2^(m * 2^(-7)) with m = 0, ..., 127. +// Lookup table for 2^(m * 2^(-6)) with m = 0, ..., 63. // Table is generated with Sollya as follow: // > display = hexadecimal; -// > for i from 0 to 127 do { D(2^(i / 128)); }; -static constexpr double EXP_M[128] = { - 0x1.0000000000000p0, 0x1.0163da9fb3335p0, 0x1.02c9a3e778061p0, - 0x1.04315e86e7f85p0, 0x1.059b0d3158574p0, 0x1.0706b29ddf6dep0, - 0x1.0874518759bc8p0, 0x1.09e3ecac6f383p0, 0x1.0b5586cf9890fp0, - 0x1.0cc922b7247f7p0, 0x1.0e3ec32d3d1a2p0, 0x1.0fb66affed31bp0, - 0x1.11301d0125b51p0, 0x1.12abdc06c31ccp0, 0x1.1429aaea92de0p0, - 0x1.15a98c8a58e51p0, 0x1.172b83c7d517bp0, 0x1.18af9388c8deap0, - 0x1.1a35beb6fcb75p0, 0x1.1bbe084045cd4p0, 0x1.1d4873168b9aap0, - 0x1.1ed5022fcd91dp0, 0x1.2063b88628cd6p0, 0x1.21f49917ddc96p0, - 0x1.2387a6e756238p0, 0x1.251ce4fb2a63fp0, 0x1.26b4565e27cddp0, - 0x1.284dfe1f56381p0, 0x1.29e9df51fdee1p0, 0x1.2b87fd0dad990p0, - 0x1.2d285a6e4030bp0, 0x1.2ecafa93e2f56p0, 0x1.306fe0a31b715p0, - 0x1.32170fc4cd831p0, 0x1.33c08b26416ffp0, 0x1.356c55f929ff1p0, - 0x1.371a7373aa9cbp0, 0x1.38cae6d05d866p0, 0x1.3a7db34e59ff7p0, - 0x1.3c32dc313a8e5p0, 0x1.3dea64c123422p0, 0x1.3fa4504ac801cp0, - 0x1.4160a21f72e2ap0, 0x1.431f5d950a897p0, 0x1.44e086061892dp0, - 0x1.46a41ed1d0057p0, 0x1.486a2b5c13cd0p0, 0x1.4a32af0d7d3dep0, - 0x1.4bfdad5362a27p0, 0x1.4dcb299fddd0dp0, 0x1.4f9b2769d2ca7p0, - 0x1.516daa2cf6642p0, 0x1.5342b569d4f82p0, 0x1.551a4ca5d920fp0, - 0x1.56f4736b527dap0, 0x1.58d12d497c7fdp0, 0x1.5ab07dd485429p0, - 0x1.5c9268a5946b7p0, 0x1.5e76f15ad2148p0, 0x1.605e1b976dc09p0, - 0x1.6247eb03a5585p0, 0x1.6434634ccc320p0, 0x1.6623882552225p0, - 0x1.68155d44ca973p0, 0x1.6a09e667f3bcdp0, 0x1.6c012750bdabfp0, - 0x1.6dfb23c651a2fp0, 0x1.6ff7df9519484p0, 0x1.71f75e8ec5f74p0, - 0x1.73f9a48a58174p0, 0x1.75feb564267c9p0, 0x1.780694fde5d3fp0, - 0x1.7a11473eb0187p0, 0x1.7c1ed0130c132p0, 0x1.7e2f336cf4e62p0, - 0x1.80427543e1a12p0, 0x1.82589994cce13p0, 0x1.8471a4623c7adp0, - 0x1.868d99b4492edp0, 0x1.88ac7d98a6699p0, 0x1.8ace5422aa0dbp0, - 0x1.8cf3216b5448cp0, 0x1.8f1ae99157736p0, 0x1.9145b0b91ffc6p0, - 0x1.93737b0cdc5e5p0, 0x1.95a44cbc8520fp0, 0x1.97d829fde4e50p0, - 0x1.9a0f170ca07bap0, 0x1.9c49182a3f090p0, 0x1.9e86319e32323p0, - 0x1.a0c667b5de565p0, 0x1.a309bec4a2d33p0, 0x1.a5503b23e255dp0, - 0x1.a799e1330b358p0, 0x1.a9e6b5579fdbfp0, 0x1.ac36bbfd3f37ap0, - 0x1.ae89f995ad3adp0, 0x1.b0e07298db666p0, 0x1.b33a2b84f15fbp0, - 0x1.b59728de5593ap0, 0x1.b7f76f2fb5e47p0, 0x1.ba5b030a1064ap0, - 0x1.bcc1e904bc1d2p0, 0x1.bf2c25bd71e09p0, 0x1.c199bdd85529cp0, - 0x1.c40ab5fffd07ap0, 0x1.c67f12e57d14bp0, 0x1.c8f6d9406e7b5p0, - 0x1.cb720dcef9069p0, 0x1.cdf0b555dc3fap0, 0x1.d072d4a07897cp0, - 0x1.d2f87080d89f2p0, 0x1.d5818dcfba487p0, 0x1.d80e316c98398p0, - 0x1.da9e603db3285p0, 0x1.dd321f301b460p0, 0x1.dfc97337b9b5fp0, - 0x1.e264614f5a129p0, 0x1.e502ee78b3ff6p0, 0x1.e7a51fbc74c83p0, - 0x1.ea4afa2a490dap0, 0x1.ecf482d8e67f1p0, 0x1.efa1bee615a27p0, - 0x1.f252b376bba97p0, 0x1.f50765b6e4540p0, 0x1.f7bfdad9cbe14p0, - 0x1.fa7c1819e90d8p0, 0x1.fd3c22b8f71f1p0, +// > for i from 0 to 63 do { D(2^(i / 64)); }; +static constexpr double EXP_M[64] = { + 0x1.0000000000000p0, 0x1.02c9a3e778061p0, 0x1.059b0d3158574p0, + 0x1.0874518759bc8p0, 0x1.0b5586cf9890fp0, 0x1.0e3ec32d3d1a2p0, + 0x1.11301d0125b51p0, 0x1.1429aaea92de0p0, 0x1.172b83c7d517bp0, + 0x1.1a35beb6fcb75p0, 0x1.1d4873168b9aap0, 0x1.2063b88628cd6p0, + 0x1.2387a6e756238p0, 0x1.26b4565e27cddp0, 0x1.29e9df51fdee1p0, + 0x1.2d285a6e4030bp0, 0x1.306fe0a31b715p0, 0x1.33c08b26416ffp0, + 0x1.371a7373aa9cbp0, 0x1.3a7db34e59ff7p0, 0x1.3dea64c123422p0, + 0x1.4160a21f72e2ap0, 0x1.44e086061892dp0, 0x1.486a2b5c13cd0p0, + 0x1.4bfdad5362a27p0, 0x1.4f9b2769d2ca7p0, 0x1.5342b569d4f82p0, + 0x1.56f4736b527dap0, 0x1.5ab07dd485429p0, 0x1.5e76f15ad2148p0, + 0x1.6247eb03a5585p0, 0x1.6623882552225p0, 0x1.6a09e667f3bcdp0, + 0x1.6dfb23c651a2fp0, 0x1.71f75e8ec5f74p0, 0x1.75feb564267c9p0, + 0x1.7a11473eb0187p0, 0x1.7e2f336cf4e62p0, 0x1.82589994cce13p0, + 0x1.868d99b4492edp0, 0x1.8ace5422aa0dbp0, 0x1.8f1ae99157736p0, + 0x1.93737b0cdc5e5p0, 0x1.97d829fde4e50p0, 0x1.9c49182a3f090p0, + 0x1.a0c667b5de565p0, 0x1.a5503b23e255dp0, 0x1.a9e6b5579fdbfp0, + 0x1.ae89f995ad3adp0, 0x1.b33a2b84f15fbp0, 0x1.b7f76f2fb5e47p0, + 0x1.bcc1e904bc1d2p0, 0x1.c199bdd85529cp0, 0x1.c67f12e57d14bp0, + 0x1.cb720dcef9069p0, 0x1.d072d4a07897cp0, 0x1.d5818dcfba487p0, + 0x1.da9e603db3285p0, 0x1.dfc97337b9b5fp0, 0x1.e502ee78b3ff6p0, + 0x1.ea4afa2a490dap0, 0x1.efa1bee615a27p0, 0x1.f50765b6e4540p0, + 0x1.fa7c1819e90d8p0, }; INLINE_FMA @@ -73,93 +52,115 @@ using FPBits = typename fputil::FPBits; FPBits xbits(x); - // When x =< -150 or nan - if (unlikely(xbits.uintval() >= 0xc316'0000U)) { - // exp(-Inf) = 0 - if (xbits.is_inf()) - return 0.0f; - // exp(nan) = nan - if (xbits.is_nan()) - return x; - if (fputil::get_round() == FE_UPWARD) - return static_cast(FPBits(FPBits::MIN_SUBNORMAL)); - if (x != 0.0f) - errno = ERANGE; - return 0.0f; - } - // x >= 128 or nan - if (unlikely(!xbits.get_sign() && (xbits.uintval() >= 0x4300'0000U))) { - if (xbits.uintval() < 0x7f80'0000U) { - int rounding = fputil::get_round(); - if (rounding == FE_DOWNWARD || rounding == FE_TOWARDZERO) - return static_cast(FPBits(FPBits::MAX_NORMAL)); + uint32_t x_u = xbits.uintval(); + uint32_t x_abs = x_u & 0x7fff'ffffU; - errno = ERANGE; - } - return x + static_cast(FPBits::inf()); - } - // |x| < 2^-25 - if (unlikely(xbits.get_unbiased_exponent() <= 101)) { - return 1.0f + x; - } // Exceptional values. - switch (xbits.uintval()) { + switch (x_u) { case 0x3b42'9d37U: // x = 0x1.853a6ep-9f if (fputil::get_round() == FE_TONEAREST) return 0x1.00870ap+0f; break; + case 0x3c02'a9adU: // x = 0x1.05535ap-7f + if (fputil::get_round() == FE_TONEAREST) + return 0x1.016b46p+0f; + break; + case 0x3ca6'6e26U: { // x = 0x1.4cdc4cp-6f + int round_mode = fputil::get_round(); + if (round_mode == FE_TONEAREST || round_mode == FE_UPWARD) + return 0x1.03a16ap+0f; + return 0x1.03a168p+0f; + } + case 0x3d92'a282U: // x = 0x1.254504p-4f + if (fputil::get_round() == FE_UPWARD) + return 0x1.0d0688p+0f; + return 0x1.0d0686p+0f; case 0xbcf3'a937U: // x = -0x1.e7526ep-6f if (fputil::get_round() == FE_TONEAREST) return 0x1.f58d62p-1f; break; + case 0xb8d3'd026U: // x = -0x1.a7a04cp-14f + if (fputil::get_round() == FE_TONEAREST) + return 0x1.fff6d2p-1f; + break; } + // // When |x| >= 128, |x| < 2^-25, or x is nan + if (unlikely(x_abs >= 0x4300'0000U || x_abs <= 0x3280'0000U)) { + // |x| < 2^-25 + if (x_abs <= 0x3280'0000U) { + return 1.0f + x; + } + // x >= 128 + if (!xbits.get_sign()) { + // x is finite + if (x_u < 0x7f80'0000U) { + int rounding = fputil::get_round(); + if (rounding == FE_DOWNWARD || rounding == FE_TOWARDZERO) + return static_cast(FPBits(FPBits::MAX_NORMAL)); + + errno = ERANGE; + } + // x is +inf or nan + return x + static_cast(FPBits::inf()); + } + // x < -150 + if (x_u >= 0xc316'0000U) { + // exp(-Inf) = 0 + if (xbits.is_inf()) + return 0.0f; + // exp(nan) = nan + if (xbits.is_nan()) + return x; + if (fputil::get_round() == FE_UPWARD) + return static_cast(FPBits(FPBits::MIN_SUBNORMAL)); + if (x != 0.0f) + errno = ERANGE; + return 0.0f; + } + } // For -150 <= x < 128, to compute 2^x, we perform the following range // reduction: find hi, mid, lo such that: // x = hi + mid + lo, in which // hi is an integer, - // mid * 2^7 is an integer - // -2^(-8) <= lo < 2^-8. + // mid * 2^6 is an integer + // -2^(-7) <= lo < 2^-7. // In particular, - // hi + mid = round(x * 2^7) * 2^(-7). + // hi + mid = round(x * 2^6) * 2^(-6). // Then, // 2^(x) = 2^(hi + mid + lo) = 2^hi * 2^mid * 2^lo. // Multiply by 2^hi is simply adding hi to the exponent field. We store - // exp(mid) in the lookup tables EXP_M. exp(lo) is computed using a degree-7 + // exp(mid) in the lookup tables EXP_M. exp(lo) is computed using a degree-4 // minimax polynomial generated by Sollya. - // x_hi = hi + mid. - int x_hi = static_cast(x * 0x1.0p7f); - // Subtract (hi + mid) from x to get lo. - x -= static_cast(x_hi) * 0x1.0p-7f; - double xd = static_cast(x); - // Make sure that -2^(-8) <= lo < 2^-8. - if (x >= 0x1.0p-8f) { - ++x_hi; - xd -= 0x1.0p-7; - } - if (x < -0x1.0p-8f) { - --x_hi; - xd += 0x1.0p-7; - } + // x_hi = round(hi + mid). + // The default rounding mode for float-to-int conversion in C++ is + // round-toward-zero. To make it round-to-nearest, we add (-1)^sign(x) * 0.5 + // before conversion. + int x_hi = + static_cast(x * 0x1.0p+6f + (xbits.get_sign() ? -0.5f : 0.5f)); // For 2-complement integers, arithmetic right shift is the same as dividing // by a power of 2 and then round down (toward negative infinity). - int hi = x_hi >> 7; - // mid = x_hi & 0x0000'007fU; - double exp_mid = EXP_M[x_hi & 0x7f]; - // Degree-6 minimax polynomial generated by Sollya with the following + int e_hi = (x_hi >> 6) + + static_cast(fputil::FloatProperties::EXPONENT_BIAS); + fputil::FPBits exp_hi( + static_cast(e_hi) + << fputil::FloatProperties::MANTISSA_WIDTH); + // mid = x_hi & 0x0000'003fU; + double exp_hi_mid = static_cast(exp_hi) * EXP_M[x_hi & 0x3f]; + // Subtract (hi + mid) from x to get lo. + x -= static_cast(x_hi) * 0x1.0p-6f; + double xd = static_cast(x); + // Degree-4 minimax polynomial generated by Sollya with the following // commands: // > display = hexadecimal; - // > Q = fpminimax((2^x - 1)/x, 5, [|D...|], [-2^-8, 2^-8]); + // > Q = fpminimax((2^x - 1)/x, 3, [|D...|], [-2^-7, 2^-7]); // > Q; double exp_lo = - fputil::polyeval(xd, 0x1p0, 0x1.62e42fefa39efp-1, 0x1.ebfbdff82c58ep-3, - 0x1.c6b08d711fe2fp-5, 0x1.3b2ab6fe3deb5p-7, - 0x1.5d72a05f45c04p-10, 0x1.4284d40c33326p-13); - fputil::FPBits result(exp_mid * exp_lo); - result.set_unbiased_exponent(static_cast( - static_cast(result.get_unbiased_exponent()) + hi)); - return static_cast(static_cast(result)); + fputil::polyeval(xd, 0x1p0, 0x1.62e42fefa2417p-1, 0x1.ebfbdff82f809p-3, + 0x1.c6b0b92131c47p-5, 0x1.3b2ab6fb568a3p-7); + double result = exp_hi_mid * exp_lo; + return static_cast(result); } } // namespace __llvm_libc diff --git a/libc/test/src/math/exp2f_test.cpp b/libc/test/src/math/exp2f_test.cpp --- a/libc/test/src/math/exp2f_test.cpp +++ b/libc/test/src/math/exp2f_test.cpp @@ -51,51 +51,29 @@ EXPECT_MATH_ERRNO(ERANGE); } -// Test with inputs which are the borders of underflow/overflow but still -// produce valid results without setting errno. -TEST(LlvmLibcExp2fTest, Borderline) { - float x; - - errno = 0; - x = float(FPBits(0x42fa0001U)); - EXPECT_MPFR_MATCH_ALL_ROUNDING(mpfr::Operation::Exp2, x, - __llvm_libc::exp2f(x), 0.5); - EXPECT_MATH_ERRNO(0); - - x = float(FPBits(0x42ffffffU)); - EXPECT_MPFR_MATCH_ALL_ROUNDING(mpfr::Operation::Exp2, x, - __llvm_libc::exp2f(x), 0.5); - EXPECT_MATH_ERRNO(0); - - x = float(FPBits(0xc2fa0001U)); - EXPECT_MPFR_MATCH_ALL_ROUNDING(mpfr::Operation::Exp2, x, - __llvm_libc::exp2f(x), 0.5); - EXPECT_MATH_ERRNO(0); - - x = float(FPBits(0xc2fc0000U)); - EXPECT_MPFR_MATCH_ALL_ROUNDING(mpfr::Operation::Exp2, x, - __llvm_libc::exp2f(x), 0.5); - EXPECT_MATH_ERRNO(0); - - x = float(FPBits(0xc2fc0001U)); - EXPECT_MPFR_MATCH_ALL_ROUNDING(mpfr::Operation::Exp2, x, - __llvm_libc::exp2f(x), 0.5); - EXPECT_MATH_ERRNO(0); - - x = float(FPBits(0xc3150000U)); - EXPECT_MPFR_MATCH_ALL_ROUNDING(mpfr::Operation::Exp2, x, - __llvm_libc::exp2f(x), 0.5); - EXPECT_MATH_ERRNO(0); - - x = float(FPBits(0x3b42'9d37U)); - EXPECT_MPFR_MATCH_ALL_ROUNDING(mpfr::Operation::Exp2, x, - __llvm_libc::exp2f(x), 0.5); - EXPECT_MATH_ERRNO(0); - - x = float(FPBits(0xbcf3'a937U)); - EXPECT_MPFR_MATCH_ALL_ROUNDING(mpfr::Operation::Exp2, x, - __llvm_libc::exp2f(x), 0.5); - EXPECT_MATH_ERRNO(0); +TEST(LlvmLibcExp2fTest, TrickyInputs) { + constexpr int N = 12; + constexpr uint32_t INPUTS[N] = { + 0x3b429d37U, /*0x1.853a6ep-9f*/ + 0x3c02a9adU, /*0x1.05535ap-7f*/ + 0x3ca66e26U, /*0x1.4cdc4cp-6f*/ + 0x3d92a282U, /*0x1.254504p-4f*/ + 0x42fa0001U, /*0x1.f40002p+6f*/ + 0x42ffffffU, /*0x1.fffffep+6f*/ + 0xb8d3d026U, /*-0x1.a7a04cp-14f*/ + 0xbcf3a937U, /*-0x1.e7526ep-6f*/ + 0xc2fa0001U, /*-0x1.f40002p+6f*/ + 0xc2fc0000U, /*-0x1.f8p+6f*/ + 0xc2fc0001U, /*-0x1.f80002p+6f*/ + 0xc3150000U, /*-0x1.2ap+7f*/ + }; + for (int i = 0; i < N; ++i) { + errno = 0; + float x = float(FPBits(INPUTS[i])); + EXPECT_MPFR_MATCH_ALL_ROUNDING(mpfr::Operation::Exp2, x, + __llvm_libc::exp2f(x), 0.5); + EXPECT_MATH_ERRNO(0); + } } TEST(LlvmLibcExp2fTest, Underflow) {