replica
/
hanchenye-llvm-project
2
0
Fork 0
Code Issues Pull Requests Projects Releases Wiki Activity

[libc] Improve the performance of expf. 

Reduce the polynomial's degree from 7 down to 4.

Currently we use a degree-7 minimax polynomial on an interval of length 2^-7
around 0 to compute `expf`. Based on the suggestion of @santoshn and the RLIBM
project (https://github.com/rutgers-apl/rlibm-all/blob/main/source/float/exp.c)
and the improvement we made with `exp2f` in https://reviews.llvm.org/D122346,
it is possible to have a good polynomial of degree-4 on a subinterval of length
2^(-7) to approximate e^x.

We did try to either reduce the degree of the polynomial down to 3 or increase
the interval size to 2^(-6), but in both cases the number of exceptional values
exploded. So we settle with using a degree-4 polynomial of the interval of
size 2^(-7) around 0.

Reviewed By: sivachandra, zimmermann6, santoshn

Differential Revision: https://reviews.llvm.org/D122418
This commit is contained in:
Tue Ly 2022-03-24 18:07:46 -04:00
parent 39d348c602
commit 6168b42225
3 changed files with 59 additions and 45 deletions
Show Stats Download Patch File Download Diff File Expand all files Collapse all files

95
libc/src/math/generic/expf.cpp
View File

@ -24,35 +24,48 @@ LLVM_LIBC_FUNCTION(float, expf, (float x)) {
using FPBits = typename fputil::FPBits<float>;
FPBits xbits(x);
// When x < log(2^-150) or nan
if (unlikely(xbits.uintval() >= 0xc2cf'f1b5U)) {
// 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<float>(FPBits(FPBits::MIN_SUBNORMAL));
errno = ERANGE;
return 0.0f;
}
// 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<float>(FPBits(FPBits::MAX_NORMAL));
uint32_t x_u = xbits.uintval();
uint32_t x_abs = x_u & 0x7fff'ffffU;
errno = ERANGE;
// Exceptional values
if (unlikely(x_u == 0xc236'bd8cU)) { // x = -0x1.6d7b18p+5f
return 0x1.108a58p-66f - x * 0x1.0p-95f;
}
// When |x| >= 89, |x| < 2^-25, or x is nan
if (unlikely(x_abs >= 0x42b2'0000U || x_abs <= 0x3280'0000U)) {
// |x| < 2^-25
if (xbits.get_unbiased_exponent() <= 101) {
return 1.0f + x;
}
return x + static_cast<float>(FPBits::inf());
}
// |x| < 2^-25
if (unlikely(xbits.get_unbiased_exponent() <= 101)) {
return 1.0f + x;
}
// When x < log(2^-150) or nan
if (xbits.uintval() >= 0xc2cf'f1b5U) {
// 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<float>(FPBits(FPBits::MIN_SUBNORMAL));
errno = ERANGE;
return 0.0f;
}
// x >= 89 or nan
if (!xbits.get_sign() && (xbits.uintval() >= 0x42b2'0000)) {
// x is finite
if (xbits.uintval() < 0x7f80'0000U) {
int rounding = fputil::get_round();
if (rounding == FE_DOWNWARD || rounding == FE_TOWARDZERO)
return static_cast<float>(FPBits(FPBits::MAX_NORMAL));
errno = ERANGE;
}
// x is +inf or nan
return x + static_cast<float>(FPBits::inf());
}
}
// For -104 < x < 89, to compute exp(x), we perform the following range
// reduction: find hi, mid, lo such that:
// x = hi + mid + lo, in which
@ -64,36 +77,30 @@ LLVM_LIBC_FUNCTION(float, expf, (float x)) {
// Then,
// exp(x) = exp(hi + mid + lo) = exp(hi) * exp(mid) * exp(lo).
// 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.
// x_hi = hi + mid.
int x_hi = static_cast<int>(x * 0x1.0p7f);
// x_hi = (hi + mid) * 2^7 = round(x * 2^7).
// 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<int>(x * 0x1.0p7f + (xbits.get_sign() ? -0.5f : 0.5f));
// Subtract (hi + mid) from x to get lo.
x -= static_cast<float>(x_hi) * 0x1.0p-7f;
double xd = static_cast<double>(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;
// mid * 2^7 = x_hi & 0x0000'007fU;
double exp_mid = EXP_M2[x_hi & 0x7f];
// 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, 0x1p0, 0x1.ffffffffff777p-1, 0x1.000000000071cp-1,
0x1.555566668e5e7p-3, 0x1.55555555ef243p-5);
return static_cast<float>(exp_hi * exp_mid * exp_lo);
}

4
libc/test/src/math/exhaustive/expf_test.cpp
View File

@ -12,6 +12,8 @@
#include "utils/MPFRWrapper/MPFRUtils.h"
#include "utils/UnitTest/FPMatcher.h"
#include <thread>
using FPBits = __llvm_libc::fputil::FPBits<float>;
namespace mpfr = __llvm_libc::testing::mpfr;
@ -32,7 +34,7 @@ struct LlvmLibcExpfExhaustiveTest : public LlvmLibcExhaustiveTest<uint32_t> {
}
};
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;

5
libc/test/src/math/expf_test.cpp
View File

@ -92,6 +92,11 @@ TEST(LlvmLibcExpfTest, Borderline) {
ASSERT_MPFR_MATCH_ALL_ROUNDING(mpfr::Operation::Exp, x, __llvm_libc::expf(x),
0.5);
EXPECT_MATH_ERRNO(0);
x = float(FPBits(0xc236bd8cU));
EXPECT_MPFR_MATCH_ALL_ROUNDING(mpfr::Operation::Exp, x, __llvm_libc::expf(x),
0.5);
EXPECT_MATH_ERRNO(0);
}
TEST(LlvmLibcExpfTest, InFloatRange) {