diff --git a/libc/src/math/generic/expm1f.cpp b/libc/src/math/generic/expm1f.cpp --- a/libc/src/math/generic/expm1f.cpp +++ b/libc/src/math/generic/expm1f.cpp @@ -24,41 +24,59 @@ using FPBits = typename fputil::FPBits; FPBits xbits(x); - // When x < log(2^-25) or nan - if (unlikely(xbits.uintval() >= 0xc18a'a123U)) { - // exp(-Inf) = 0 - if (xbits.is_inf()) - return -1.0f; - // exp(nan) = nan - if (xbits.is_nan()) - return x; + uint32_t x_u = xbits.uintval(); + uint32_t x_abs = x_u & 0x7fff'ffffU; + + // Exceptional value + if (unlikely(x_u == 0x3e35'bec5U)) { // x = 0x1.6b7d8ap-3f int round_mode = fputil::get_round(); - if (round_mode == FE_UPWARD || round_mode == FE_TOWARDZERO) - return -0x1.ffff'fep-1f; // -1.0f + 0x1.0p-24f - return -1.0f; + if (round_mode == FE_TONEAREST || round_mode == FE_UPWARD) + return 0x1.8dbe64p-3f; + return 0x1.8dbe62p-3f; } - // x >= 89 or nan - if (unlikely(!xbits.get_sign() && (xbits.uintval() >= 0x42b2'0000))) { - if (xbits.uintval() < 0x7f80'0000U) { - int rounding = fputil::get_round(); - if (rounding == FE_DOWNWARD || rounding == FE_TOWARDZERO) - return static_cast(FPBits(FPBits::MAX_NORMAL)); - errno = ERANGE; + // When |x| > 25*log(2), or nan + if (unlikely(x_abs >= 0x418a'a123U)) { + // x < log(2^-25) + if (xbits.get_sign()) { + // exp(-Inf) = 0 + if (xbits.is_inf()) + return -1.0f; + // exp(nan) = nan + if (xbits.is_nan()) + return x; + int round_mode = fputil::get_round(); + if (round_mode == FE_UPWARD || round_mode == FE_TOWARDZERO) + return -0x1.ffff'fep-1f; // -1.0f + 0x1.0p-24f + return -1.0f; + } else { + // x >= 89 or nan + if (xbits.uintval() >= 0x42b2'0000) { + if (xbits.uintval() < 0x7f80'0000U) { + int rounding = fputil::get_round(); + if (rounding == FE_DOWNWARD || rounding == FE_TOWARDZERO) + return static_cast(FPBits(FPBits::MAX_NORMAL)); + + errno = ERANGE; + } + return x + static_cast(FPBits::inf()); + } } - return x + static_cast(FPBits::inf()); } - int unbiased_exponent = static_cast(xbits.get_unbiased_exponent()); // |x| < 2^-4 - if (unbiased_exponent < 123) { + if (x_abs < 0x3d80'0000U) { // |x| < 2^-25 - if (unbiased_exponent < 102) { + if (x_abs < 0x3300'0000U) { // x = -0.0f if (unlikely(xbits.uintval() == 0x8000'0000U)) return x; - // When |x| < 2^-25, the relative error: - // |(e^x - 1) - x| / |x| < |x^2| / |x| = |x| < 2^-25 < epsilon(1)/2. + // When |x| < 2^-25, the relative error of the approximation e^x - 1 ~ x + // is: + // |(e^x - 1) - x| / |e^x - 1| < |x^2| / |x| + // = |x| + // < 2^-25 + // < epsilon(1)/2. // So the correctly rounded values of expm1(x) are: // = x + eps(x) if rounding mode = FE_UPWARD, // or (rounding mode = FE_TOWARDZERO and x is negative), @@ -67,20 +85,21 @@ // fma(x, x, x) ~ x + x^2 instead. return fputil::fma(x, x, x); } + // 2^-25 <= |x| < 2^-4 double xd = static_cast(x); double xsq = xd * xd; // Degree-8 minimax polynomial generated by Sollya with: // > display = hexadecimal; - // > P = fpminimax(expm1(x)/x, 7, [|D...|], [-2^-4, 2^-4]); + // > P = fpminimax((expm1(x) - x)/x^2, 6, [|D...|], [-2^-4, 2^-4]); double r = - fputil::polyeval(xd, 0x1p-1, 0x1.55555555559abp-3, 0x1.55555555551a7p-5, - 0x1.111110f70f2a4p-7, 0x1.6c16c17639e82p-10, - 0x1.a02526febbea6p-13, 0x1.a01dc40888fcdp-16); + fputil::polyeval(xd, 0x1p-1, 0x1.55555555557ddp-3, 0x1.55555555552fap-5, + 0x1.111110fcd58b7p-7, 0x1.6c16c1717660bp-10, + 0x1.a0241f0006d62p-13, 0x1.a01e3f8d3c06p-16); return static_cast(fputil::fma(r, xsq, xd)); } - // For -18 < x < 89, to compute exp(x), we perform the following range + // For -18 < x < 89, to compute expm1(x), we perform the following range // reduction: find hi, mid, lo such that: // x = hi + mid + lo, in which // hi is an integer, @@ -89,48 +108,30 @@ // In particular, // hi + mid = round(x * 2^7) * 2^(-7). // Then, - // exp(x) = exp(hi + mid + lo) = exp(hi) * exp(mid) * exp(lo). + // expm1(x) = exp(hi + mid + lo) - 1 = exp(hi) * exp(mid) * exp(lo) - 1. // We store exp(hi) and exp(mid) in the lookup tables EXP_M1 and EXP_M2 - // respectively. exp(lo) is computed using a degree-7 minimax polynomial + // respectively. exp(lo) is computed using a degree-4 minimax polynomial // generated by Sollya. - // Exceptional value - if (xbits.uintval() == 0xbdc1'c6cbU) { - // x = -0x1.838d96p-4f - int round_mode = fputil::get_round(); - if (round_mode == FE_TONEAREST || round_mode == FE_DOWNWARD) - return -0x1.71c884p-4f; - return -0x1.71c882p-4f; - } - // x_hi = hi + mid. - int x_hi = static_cast(x * 0x1.0p7f); + int x_hi = static_cast(x * 0x1.0p7f + (xbits.get_sign() ? -0.5f : 0.5f)); // 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 += 104 << 7; // hi = x_hi >> 7 double exp_hi = EXP_M1[x_hi >> 7]; // lo = x_hi & 0x0000'007fU; double exp_mid = EXP_M2[x_hi & 0x7f]; double exp_hi_mid = exp_hi * exp_mid; - // Degree-7 minimax polynomial generated by Sollya with the following + // Degree-4 minimax polynomial generated by Sollya with the following // commands: // > display = hexadecimal; - // > Q = fpminimax(expm1(x)/x, 6, [|D...|], [-2^-8, 2^-8]); + // > Q = fpminimax(expm1(x)/x, 3, [|D...|], [-2^-8, 2^-8]); // > Q; - double exp_lo = fputil::polyeval( - xd, 0x1p0, 0x1p0, 0x1p-1, 0x1.5555555555555p-3, 0x1.55555555553ap-5, - 0x1.1111111204dfcp-7, 0x1.6c16cb2da593ap-10, 0x1.9ff1648996d2ep-13); + double exp_lo = + fputil::polyeval(xd, 0x1.0p0, 0x1.ffffffffff777p-1, 0x1.000000000071cp-1, + 0x1.555566668e5e7p-3, 0x1.55555555ef243p-5); return static_cast(fputil::fma(exp_hi_mid, exp_lo, -1.0)); } diff --git a/libc/test/src/math/exhaustive/exhaustive_test.cpp b/libc/test/src/math/exhaustive/exhaustive_test.cpp --- a/libc/test/src/math/exhaustive/exhaustive_test.cpp +++ b/libc/test/src/math/exhaustive/exhaustive_test.cpp @@ -15,6 +15,7 @@ #include #include "exhaustive_test.h" +#include "src/__support/FPUtil/FPBits.h" #include "utils/UnitTest/Test.h" template @@ -28,14 +29,26 @@ thread_list.emplace_back([this, begin, end, rounding]() { std::stringstream msg; msg << "-- Testing from " << begin << " to " << end << " [0x" << std::hex - << begin << ", 0x" << end << ") ..." << std::endl; + << begin << ", 0x" << end << "), [" << std::hexfloat + << float(__llvm_libc::fputil::FPBits( + static_cast(begin))) + << ", " + << float( + __llvm_libc::fputil::FPBits(static_cast(end))) + << ") ..." << std::endl; std::cout << msg.str(); msg.str(""); bool result = check(begin, end, rounding); msg << "** Finished testing from " << std::dec << begin << " to " << end - << " [0x" << std::hex << begin << ", 0x" << end + << " [0x" << std::hex << begin << ", 0x" << end << "), [" + << std::hexfloat + << float(__llvm_libc::fputil::FPBits( + static_cast(begin))) + << ", " + << float( + __llvm_libc::fputil::FPBits(static_cast(end))) << ") : " << (result ? "PASSED" : "FAILED") << std::endl; std::cout << msg.str(); }); diff --git a/libc/test/src/math/exhaustive/expm1f_test.cpp b/libc/test/src/math/exhaustive/expm1f_test.cpp --- a/libc/test/src/math/exhaustive/expm1f_test.cpp +++ b/libc/test/src/math/exhaustive/expm1f_test.cpp @@ -12,6 +12,8 @@ #include "utils/MPFRWrapper/MPFRUtils.h" #include "utils/UnitTest/FPMatcher.h" +#include + using FPBits = __llvm_libc::fputil::FPBits; namespace mpfr = __llvm_libc::testing::mpfr; @@ -20,18 +22,19 @@ bool check(uint32_t start, uint32_t stop, mpfr::RoundingMode rounding) override { mpfr::ForceRoundingMode r(rounding); - uint32_t bits = start; + uint32_t bits = stop; bool result = true; do { FPBits xbits(bits); float x = float(xbits); result &= EXPECT_MPFR_MATCH(mpfr::Operation::Expm1, x, __llvm_libc::expm1f(x), 0.5, rounding); - } while (bits++ < stop); + } while (bits-- > start); + return result; } }; -static constexpr int NUM_THREADS = 16; +static const int NUM_THREADS = std::thread::hardware_concurrency(); // Range: [0, 89]; static constexpr uint32_t POS_START = 0x0000'0000U; diff --git a/libc/test/src/math/expm1f_test.cpp b/libc/test/src/math/expm1f_test.cpp --- a/libc/test/src/math/expm1f_test.cpp +++ b/libc/test/src/math/expm1f_test.cpp @@ -92,6 +92,11 @@ ASSERT_MPFR_MATCH_ALL_ROUNDING(mpfr::Operation::Expm1, x, __llvm_libc::expm1f(x), 0.5); EXPECT_MATH_ERRNO(0); + + x = float(FPBits(0x3e35bec5U)); + ASSERT_MPFR_MATCH_ALL_ROUNDING(mpfr::Operation::Expm1, x, + __llvm_libc::expm1f(x), 0.5); + EXPECT_MATH_ERRNO(0); } TEST(LlvmLibcExpm1fTest, InFloatRange) {