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-rw-r--r--libc/src/math/generic/exp2.cpp398
1 files changed, 2 insertions, 396 deletions
diff --git a/libc/src/math/generic/exp2.cpp b/libc/src/math/generic/exp2.cpp
index 154154f..20e1ff5 100644
--- a/libc/src/math/generic/exp2.cpp
+++ b/libc/src/math/generic/exp2.cpp
@@ -7,404 +7,10 @@
//===----------------------------------------------------------------------===//
#include "src/math/exp2.h"
-#include "common_constants.h" // Lookup tables EXP2_MID1 and EXP_M2.
-#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_utils.h" // ziv_test_denorm.
+#include "src/__support/math/exp2.h"
namespace LIBC_NAMESPACE_DECL {
-using fputil::DoubleDouble;
-using fputil::TripleDouble;
-using Float128 = typename fputil::DyadicFloat<128>;
-
-using LIBC_NAMESPACE::operator""_u128;
-
-// Error bounds:
-// Errors when using double precision.
-#ifdef LIBC_TARGET_CPU_HAS_FMA_DOUBLE
-constexpr double ERR_D = 0x1.0p-63;
-#else
-constexpr double ERR_D = 0x1.8p-63;
-#endif // LIBC_TARGET_CPU_HAS_FMA_DOUBLE
-
-#ifndef LIBC_MATH_HAS_SKIP_ACCURATE_PASS
-// Errors when using double-double precision.
-constexpr double ERR_DD = 0x1.0p-100;
-#endif // LIBC_MATH_HAS_SKIP_ACCURATE_PASS
-
-namespace {
-
-// 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;
-}
-
-#ifndef LIBC_MATH_HAS_SKIP_ACCURATE_PASS
-// 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) {
- constexpr Float128 COEFFS_128[]{
- {Sign::POS, -127, 0x80000000'00000000'00000000'00000000_u128}, // 1.0
- {Sign::POS, -128, 0xb17217f7'd1cf79ab'c9e3b398'03f2f6af_u128},
- {Sign::POS, -128, 0x3d7f7bff'058b1d50'de2d60dd'9c9a1d9f_u128},
- {Sign::POS, -132, 0xe35846b8'2505fc59'9d3b15d9'e7fb6897_u128},
- {Sign::POS, -134, 0x9d955b7d'd273b94e'184462f6'bcd2b9e7_u128},
- {Sign::POS, -137, 0xaec3ff3c'53398883'39ea1bb9'64c51a89_u128},
- {Sign::POS, -138, 0x2861225f'345c396a'842c5341'8fa8ae61_u128},
- {Sign::POS, -144, 0xffe5fe2d'109a319d'7abeb5ab'd5ad2079_u128},
- };
-
- 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 2^(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;
-}
-#endif // LIBC_MATH_HAS_SKIP_ACCURATE_PASS
-
-// 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);
-
-#ifdef LIBC_MATH_HAS_SKIP_ACCURATE_PASS
- return ziv_test_denorm</*SKIP_ZIV_TEST=*/true>(hi, exp_mid.hi, lo, ERR_D)
- .value();
-#else
- 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);
-#endif // LIBC_MATH_HAS_SKIP_ACCURATE_PASS
-}
-
-// Check for exceptional cases when:
-// * log2(1 - 2^-54) < x < log2(1 + 2^-53)
-// * x >= 1024
-// * x <= -1022
-// * x is inf or nan
-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| < log2(1 + 2^-53)
- if (x_abs <= 0x3ca71547652b82fd) {
- // 2^(x) ~ 1 + x/2
- return fputil::multiply_add(x, 0.5, 1.0);
- }
-
- // x <= -1022 || x >= 1024 or inf/nan.
- if (x_u > 0xc08ff00000000000) {
- // x <= -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 FPBits::min_subnormal().get_val();
- 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 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, exp2, (double x)) {
- using FPBits = typename fputil::FPBits<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);
-
-#ifdef LIBC_MATH_HAS_SKIP_ACCURATE_PASS
- // 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
- 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 = 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) << 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 = exp2_f128(dx, hi, idx1, idx2);
-
- return static_cast<double>(r_f128);
-#endif // LIBC_MATH_HAS_SKIP_ACCURATE_PASS
-}
+LLVM_LIBC_FUNCTION(double, exp2, (double x)) { return math::exp2(x); }
} // namespace LIBC_NAMESPACE_DECL
generated by cgit v1.1 at 2025-10-05 12:14:07 +0000