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-rw-r--r--libc/src/math/generic/exp.cpp429
1 files changed, 2 insertions, 427 deletions
diff --git a/libc/src/math/generic/exp.cpp b/libc/src/math/generic/exp.cpp
index 143800ca078a..dc4d2ca480cb 100644
--- a/libc/src/math/generic/exp.cpp
+++ b/libc/src/math/generic/exp.cpp
@@ -7,434 +7,9 @@
//===----------------------------------------------------------------------===//
#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"
-#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/integer_literals.h"
-#include "src/__support/macros/config.h"
-#include "src/__support/macros/optimization.h" // LIBC_UNLIKELY
-
+#include "src/__support/math/exp.h"
namespace LIBC_NAMESPACE_DECL {
-using fputil::DoubleDouble;
-using fputil::TripleDouble;
-using Float128 = typename fputil::DyadicFloat<128>;
-
-using LIBC_NAMESPACE::operator""_u128;
-
-// log2(e)
-constexpr double LOG2_E = 0x1.71547652b82fep+0;
-
-// Error bounds:
-// Errors when using double precision.
-constexpr double ERR_D = 0x1.8p-63;
-
-#ifndef LIBC_MATH_HAS_SKIP_ACCURATE_PASS
-// Errors when using double-double precision.
-constexpr double ERR_DD = 0x1.0p-99;
-#endif // LIBC_MATH_HAS_SKIP_ACCURATE_PASS
-
-// -2^-12 * log(2)
-// > a = -2^-12 * log(2);
-// > b = round(a, 30, RN);
-// > c = round(a - b, 30, RN);
-// > d = round(a - b - c, D, RN);
-// Errors < 1.5 * 2^-133
-constexpr double MLOG_2_EXP2_M12_HI = -0x1.62e42ffp-13;
-constexpr double MLOG_2_EXP2_M12_MID = 0x1.718432a1b0e26p-47;
-
-#ifndef LIBC_MATH_HAS_SKIP_ACCURATE_PASS
-constexpr double MLOG_2_EXP2_M12_MID_30 = 0x1.718432ap-47;
-constexpr double MLOG_2_EXP2_M12_LO = 0x1.b0e2633fe0685p-79;
-#endif // LIBC_MATH_HAS_SKIP_ACCURATE_PASS
-
-namespace {
-
-// Polynomial approximations with double precision:
-// Return expm1(dx) / x ~ 1 + dx / 2 + dx^2 / 6 + dx^3 / 24.
-// For |dx| < 2^-13 + 2^-30:
-// | output - expm1(dx) / dx | < 2^-51.
-LIBC_INLINE double poly_approx_d(double dx) {
- // dx^2
- double dx2 = dx * dx;
- // c0 = 1 + dx / 2
- double c0 = fputil::multiply_add(dx, 0.5, 1.0);
- // c1 = 1/6 + dx / 24
- double c1 =
- fputil::multiply_add(dx, 0x1.5555555555555p-5, 0x1.5555555555555p-3);
- // p = dx^2 * c1 + c0 = 1 + dx / 2 + dx^2 / 6 + dx^3 / 24
- double p = fputil::multiply_add(dx2, c1, c0);
- return p;
-}
-
-#ifndef LIBC_MATH_HAS_SKIP_ACCURATE_PASS
-// Polynomial approximation with double-double precision:
-// Return exp(dx) ~ 1 + dx + dx^2 / 2 + ... + dx^6 / 720
-// For |dx| < 2^-13 + 2^-30:
-// | output - exp(dx) | < 2^-101
-DoubleDouble poly_approx_dd(const DoubleDouble &dx) {
- // Taylor polynomial.
- constexpr DoubleDouble COEFFS[] = {
- {0, 0x1p0}, // 1
- {0, 0x1p0}, // 1
- {0, 0x1p-1}, // 1/2
- {0x1.5555555555555p-57, 0x1.5555555555555p-3}, // 1/6
- {0x1.5555555555555p-59, 0x1.5555555555555p-5}, // 1/24
- {0x1.1111111111111p-63, 0x1.1111111111111p-7}, // 1/120
- {-0x1.f49f49f49f49fp-65, 0x1.6c16c16c16c17p-10}, // 1/720
- };
-
- 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 + dx + dx^2 / 2 + ... + dx^7 / 5040
-// For |dx| < 2^-13 + 2^-30:
-// | output - exp(dx) | < 2^-126.
-Float128 poly_approx_f128(const Float128 &dx) {
- constexpr Float128 COEFFS_128[]{
- {Sign::POS, -127, 0x80000000'00000000'00000000'00000000_u128}, // 1.0
- {Sign::POS, -127, 0x80000000'00000000'00000000'00000000_u128}, // 1.0
- {Sign::POS, -128, 0x80000000'00000000'00000000'00000000_u128}, // 0.5
- {Sign::POS, -130, 0xaaaaaaaa'aaaaaaaa'aaaaaaaa'aaaaaaab_u128}, // 1/6
- {Sign::POS, -132, 0xaaaaaaaa'aaaaaaaa'aaaaaaaa'aaaaaaab_u128}, // 1/24
- {Sign::POS, -134, 0x88888888'88888888'88888888'88888889_u128}, // 1/120
- {Sign::POS, -137, 0xb60b60b6'0b60b60b'60b60b60'b60b60b6_u128}, // 1/720
- {Sign::POS, -140, 0xd00d00d0'0d00d00d'00d00d00'd00d00d0_u128}, // 1/5040
- };
-
- 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 exp_f128(double x, double kd, int idx1, int idx2) {
- // Recalculate dx:
-
- double t1 = fputil::multiply_add(kd, MLOG_2_EXP2_M12_HI, x); // exact
- double t2 = kd * MLOG_2_EXP2_M12_MID_30; // exact
- double t3 = kd * MLOG_2_EXP2_M12_LO; // Error < 2^-133
-
- Float128 dx = fputil::quick_add(
- Float128(t1), fputil::quick_add(Float128(t2), Float128(t3)));
-
- // 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 += static_cast<int>(kd) >> 12;
-
- return r;
-}
-
-// Compute exp(x) with double-double precision.
-DoubleDouble exp_double_double(double x, double kd,
- const DoubleDouble &exp_mid) {
- // Recalculate dx:
- // dx = x - k * 2^-12 * log(2)
- double t1 = fputil::multiply_add(kd, MLOG_2_EXP2_M12_HI, x); // exact
- double t2 = kd * MLOG_2_EXP2_M12_MID_30; // exact
- double t3 = kd * MLOG_2_EXP2_M12_LO; // Error < 2^-130
-
- DoubleDouble dx = fputil::exact_add(t1, t2);
- dx.lo += t3;
-
- // Degree-6 Taylor polynomial approximation in double-double precision.
- // | p - exp(x) | < 2^-100.
- DoubleDouble p = poly_approx_dd(dx);
-
- // Error bounds: 2^-99.
- DoubleDouble r = fputil::quick_mult(exp_mid, p);
-
- return r;
-}
-#endif // LIBC_MATH_HAS_SKIP_ACCURATE_PASS
-
-// Check for exceptional cases when
-// |x| <= 2^-53 or x < log(2^-1075) or x >= 0x1.6232bdd7abcd3p+9
-double set_exceptional(double x) {
- using FPBits = typename fputil::FPBits<double>;
- FPBits xbits(x);
-
- uint64_t x_u = xbits.uintval();
- uint64_t x_abs = xbits.abs().uintval();
-
- // |x| <= 2^-53
- if (x_abs <= 0x3ca0'0000'0000'0000ULL) {
- // exp(x) ~ 1 + x
- return 1 + x;
- }
-
- // x <= log(2^-1075) || x >= 0x1.6232bdd7abcd3p+9 or inf/nan.
-
- // x <= log(2^-1075) or -inf/nan
- if (x_u >= 0xc087'4910'd52d'3052ULL) {
- // 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 FPBits::min_subnormal().get_val();
- fputil::set_errno_if_required(ERANGE);
- fputil::raise_except_if_required(FE_UNDERFLOW);
- return 0.0;
- }
-
- // x >= round(log(MAX_NORMAL), D, RU) = 0x1.62e42fefa39fp+9 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 FPBits::max_normal().get_val();
-
- fputil::set_errno_if_required(ERANGE);
- fputil::raise_except_if_required(FE_OVERFLOW);
- }
- // x is +inf or nan
- return x + FPBits::inf().get_val();
-}
-
-} // namespace
-
-LLVM_LIBC_FUNCTION(double, exp, (double x)) {
- using FPBits = typename fputil::FPBits<double>;
- FPBits xbits(x);
-
- uint64_t x_u = xbits.uintval();
-
- // Upper bound: max normal number = 2^1023 * (2 - 2^-52)
- // > round(log (2^1023 ( 2 - 2^-52 )), D, RU) = 0x1.62e42fefa39fp+9
- // > round(log (2^1023 ( 2 - 2^-52 )), D, RD) = 0x1.62e42fefa39efp+9
- // > round(log (2^1023 ( 2 - 2^-52 )), D, RN) = 0x1.62e42fefa39efp+9
- // > round(exp(0x1.62e42fefa39fp+9), D, RN) = infty
-
- // Lower bound: min denormal number / 2 = 2^-1075
- // > round(log(2^-1075), D, RN) = -0x1.74910d52d3052p9
-
- // Another lower bound: min normal number = 2^-1022
- // > round(log(2^-1022), D, RN) = -0x1.6232bdd7abcd2p9
-
- // x < log(2^-1075) or x >= 0x1.6232bdd7abcd3p+9 or |x| < 2^-53.
- if (LIBC_UNLIKELY(x_u >= 0xc0874910d52d3052 ||
- (x_u < 0xbca0000000000000 && x_u >= 0x40862e42fefa39f0) ||
- x_u < 0x3ca0000000000000)) {
- return set_exceptional(x);
- }
-
- // 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
- // in which:
- // hi is an integer
- // mid1 * 2^6 is an integer
- // mid2 * 2^12 is an integer
- // then:
- // exp(x) = 2^hi * 2^(mid1) * 2^(mid2) * exp(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.
- // - exp(lo) ~ 1 + lo + a0 * lo^2 + ...
- //
- // They can be defined by:
- // hi + mid1 + mid2 = 2^(-12) * round(2^12 * log_2(e) * x)
- // If we store L2E = round(log2(e), D, RN), then:
- // log2(e) - L2E ~ 1.5 * 2^(-56)
- // So the errors when computing in double precision is:
- // | x * 2^12 * log_2(e) - D(x * 2^12 * L2E) | <=
- // <= | x * 2^12 * log_2(e) - x * 2^12 * L2E | +
- // + | x * 2^12 * L2E - D(x * 2^12 * L2E) |
- // <= 2^12 * ( |x| * 1.5 * 2^-56 + eps(x)) for RN
- // 2^12 * ( |x| * 1.5 * 2^-56 + 2*eps(x)) for other rounding modes.
- // So if:
- // hi + mid1 + mid2 = 2^(-12) * round(x * 2^12 * L2E) is computed entirely
- // in double precision, the reduced argument:
- // lo = x - log(2) * (hi + mid1 + mid2) is bounded by:
- // |lo| <= 2^-13 + (|x| * 1.5 * 2^-56 + 2*eps(x))
- // < 2^-13 + (1.5 * 2^9 * 1.5 * 2^-56 + 2*2^(9 - 52))
- // < 2^-13 + 2^-41
- //
-
- // The following trick computes the round(x * L2E) more efficiently
- // than using the rounding instructions, with the tradeoff for less accuracy,
- // and hence a slightly larger range for the reduced argument `lo`.
- //
- // To be precise, since |x| < |log(2^-1075)| < 1.5 * 2^9,
- // |x * 2^12 * L2E| < 1.5 * 2^9 * 1.5 < 2^23,
- // So we can fit the rounded result round(x * 2^12 * L2E) 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 * L2E).
- //
- // 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*L2E.
- //
- // One small problem with this approach is that the sum (x*2^12*L2E + 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 * L2E + C,
- // where C = 2^33 + 2^32 + 2^-1, then if
- // k = int32_t(lower 51 bits of double(x * 2^12 * L2E + C) >> 19),
- // the reduced argument:
- // lo = x - log(2) * 2^-12 * k is bounded by:
- // |lo| <= 2^-13 + 2^-41 + 2^-12*2^-19
- // = 2^-13 + 2^-31 + 2^-41.
- //
- // Finally, notice that k only uses the mantissa of x * 2^12 * L2E, 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 * L2E + C) >> 19).
-
- // Rounding errors <= 2^-31 + 2^-41.
- double tmp = fputil::multiply_add(x, LOG2_E, 0x1.8000'0000'4p21);
- int k = static_cast<int>(cpp::bit_cast<uint64_t>(tmp) >> 19);
- double kd = static_cast<double>(k);
-
- uint32_t idx1 = (k >> 6) & 0x3f;
- uint32_t idx2 = k & 0x3f;
- int hi = k >> 12;
-
- bool denorm = (hi <= -1022);
-
- 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);
-
- // |x - (hi + mid1 + mid2) * log(2) - dx| < 2^11 * eps(M_LOG_2_EXP2_M12.lo)
- // = 2^11 * 2^-13 * 2^-52
- // = 2^-54.
- // |dx| < 2^-13 + 2^-30.
- double lo_h = fputil::multiply_add(kd, MLOG_2_EXP2_M12_HI, x); // exact
- double dx = fputil::multiply_add(kd, MLOG_2_EXP2_M12_MID, lo_h);
-
- // We use the degree-4 Taylor polynomial to approximate exp(lo):
- // exp(lo) ~ 1 + lo + lo^2 / 2 + lo^3 / 6 + lo^4 / 24 = 1 + lo * P(lo)
- // So that the errors are bounded by:
- // |P(lo) - expm1(lo)/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_(dx) - P(dx)| < 2^-51
- // => |dx * P_(dx) - expm1(lo) | < 1.5 * 2^-64
- // => 2^(mid1 + mid2) * |dx * P_(dx) - expm1(lo)| < 1.5 * 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 1.5 * 2^-63.
-
- double mid_lo = dx * exp_mid.hi;
-
- // Approximate expm1(dx)/dx ~ 1 + dx / 2 + dx^2 / 6 + dx^3 / 24.
- double p = poly_approx_d(dx);
-
- double lo = fputil::multiply_add(p, mid_lo, exp_mid.lo);
-
-#ifdef LIBC_MATH_HAS_SKIP_ACCURATE_PASS
- if (LIBC_UNLIKELY(denorm)) {
- return ziv_test_denorm</*SKIP_ZIV_TEST=*/true>(hi, exp_mid.hi, lo, ERR_D)
- .value();
- } else {
- // 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) << FPBits::FRACTION_LEN;
- double r =
- cpp::bit_cast<double>(exp_hi + cpp::bit_cast<int64_t>(exp_mid.hi + lo));
- return r;
- }
-#else
- if (LIBC_UNLIKELY(denorm)) {
- if (auto r = ziv_test_denorm(hi, exp_mid.hi, lo, ERR_D);
- LIBC_LIKELY(r.has_value()))
- return r.value();
- } else {
- 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) << FPBits::FRACTION_LEN;
- double r = cpp::bit_cast<double>(exp_hi + cpp::bit_cast<int64_t>(upper));
- return r;
- }
- }
-
- // Use double-double
- DoubleDouble r_dd = exp_double_double(x, kd, exp_mid);
-
- if (LIBC_UNLIKELY(denorm)) {
- if (auto r = ziv_test_denorm(hi, r_dd.hi, r_dd.lo, ERR_DD);
- LIBC_LIKELY(r.has_value()))
- return r.value();
- } else {
- 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)) {
- int64_t exp_hi = static_cast<int64_t>(hi) << FPBits::FRACTION_LEN;
- double r =
- cpp::bit_cast<double>(exp_hi + cpp::bit_cast<int64_t>(upper_dd));
- return r;
- }
- }
-
- // Use 128-bit precision
- Float128 r_f128 = exp_f128(x, kd, idx1, idx2);
-
- return static_cast<double>(r_f128);
-#endif // LIBC_MATH_HAS_SKIP_ACCURATE_PASS
-}
+LLVM_LIBC_FUNCTION(double, exp, (double x)) { return math::exp(x); }
} // namespace LIBC_NAMESPACE_DECL
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