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[libc][math] Simplify tanf implementation and improve its performance. 

Simplify `tanf` implementation and improve its performance.

Completely reuse the implementation of `sinf`, `cosf`, `sincosf` and use
the definition `tan(x) = sin(x)/cos(x)`.

Performance benchmark using perf tool from the CORE-MATH project on Ryzen 1700:
```
$ CORE_MATH_PERF_MODE="rdtsc" ./perf.sh tanf
GNU libc version: 2.35
GNU libc release: stable
CORE-MATH reciprocal throughput   : 18.558
System LIBC reciprocal throughput : 49.919

BEFORE:
LIBC reciprocal throughput        : 36.480
LIBC reciprocal throughput        : 27.217    (with `-msse4.2` flag)
LIBC reciprocal throughput        : 20.205    (with `-mfma` flag)

AFTER:
LIBC reciprocal throughput        : 30.337
LIBC reciprocal throughput        : 21.072    (with `-msse4.2` flag)
LIBC reciprocal throughput        : 15.804    (with `-mfma` flag)

$ CORE_MATH_PERF_MODE="rdtsc" ./perf.sh tanf --latency
GNU libc version: 2.35
GNU libc release: stable
CORE-MATH latency   : 56.702
System LIBC latency : 107.206

BEFORE
LIBC latency        : 97.598
LIBC latency        : 91.119   (with `-msse4.2` flag)
LIBC latency        : 82.655    (with `-mfma` flag)

AFTER
LIBC latency        : 74.560
LIBC latency        : 66.575    (with `-msse4.2` flag)
LIBC latency        : 61.636    (with `-mfma` flag)
```

Reviewed By: zimmermann6

Differential Revision: https://reviews.llvm.org/D134575
This commit is contained in:
Tue Ly 2022-09-23 19:12:38 -04:00
parent f3e02989e6
commit e15b2da42f
4 changed files with 53 additions and 139 deletions
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2
libc/docs/math.rst
View File

@ -251,7 +251,7 @@ Performance
+--------------+-----------+-------------------+-----------+-------------------+-------------------------------------+------------+-------------------------+--------------+---------------+
| sinhf | 13 | 63 | 48 | 137 | :math:`[-10, 10]` | Ryzen 1700 | Ubuntu 22.04 LTS x86_64 | Clang 14.0.0 | FMA |
+--------------+-----------+-------------------+-----------+-------------------+-------------------------------------+------------+-------------------------+--------------+---------------+
| tanf | 19 | 50 | 82 | 107 | :math:`[-\pi, \pi]` | Ryzen 1700 | Ubuntu 20.04 LTS x86_64 | Clang 12.0.0 | FMA |
| tanf | 16 | 50 | 61 | 107 | :math:`[-\pi, \pi]` | Ryzen 1700 | Ubuntu 22.04 LTS x86_64 | Clang 14.0.0 | FMA |
+--------------+-----------+-------------------+-----------+-------------------+-------------------------------------+------------+-------------------------+--------------+---------------+
| tanhf | 13 | 55 | 57 | 123 | :math:`[-10, 10]` | Ryzen 1700 | Ubuntu 22.04 LTS x86_64 | Clang 14.0.0 | FMA |
+--------------+-----------+-------------------+-----------+-------------------+-------------------------------------+------------+-------------------------+--------------+---------------+

1
libc/src/math/generic/CMakeLists.txt
View File

@ -138,6 +138,7 @@ add_entrypoint_object(
../tanf.h
DEPENDS
.range_reduction
.sincosf_utils
libc.include.math
libc.src.errno.errno
libc.src.__support.FPUtil.fenv_impl

181
libc/src/math/generic/tanf.cpp
View File

@ -7,6 +7,7 @@
//===----------------------------------------------------------------------===//
#include "src/math/tanf.h"
#include "sincosf_utils.h"
#include "src/__support/FPUtil/FEnvImpl.h"
#include "src/__support/FPUtil/FPBits.h"
#include "src/__support/FPUtil/PolyEval.h"
@ -17,115 +18,32 @@
#include <errno.h>
#if defined(LIBC_TARGET_HAS_FMA)
#include "range_reduction_fma.h"
// using namespace __llvm_libc::fma;
using __llvm_libc::fma::FAST_PASS_BOUND;
using __llvm_libc::fma::large_range_reduction;
using __llvm_libc::fma::small_range_reduction;
#else
#include "range_reduction.h"
// using namespace __llvm_libc::generic;
using __llvm_libc::generic::FAST_PASS_BOUND;
using __llvm_libc::generic::large_range_reduction;
using __llvm_libc::generic::small_range_reduction;
#endif
namespace __llvm_libc {
// Lookup table for tan(k * pi/32) with k = -15..15 organized as follow:
// TAN_K_OVER_32[k] = tan(k * pi/32) for k = 0..15
// TAN_K_OVER_32[k] = tan((k - 31) * pi/32) for k = 16..31.
// This organization allows us to simply do the lookup:
// TAN_K_OVER_32[k & 31] for k of type int(32/64) with 2-complement
// representation.
// The values of tan(k * pi/32) are generated by Sollya with:
// for k from 0 -15 to 15 do { round(tan(k*pi/32), D, RN); };
static constexpr double TAN_K_PI_OVER_32[32] = {
0.0000000000000000, 0x1.936bb8c5b2da2p-4, 0x1.975f5e0553158p-3,
0x1.36a08355c63dcp-2, 0x1.a827999fcef32p-2, 0x1.11ab7190834ecp-1,
0x1.561b82ab7f99p-1, 0x1.a43002ae4285p-1, 0x1.0000000000000p0,
0x1.37efd8d87607ep0, 0x1.7f218e25a7461p0, 0x1.def13b73c1406p0,
0x1.3504f333f9de6p1, 0x1.a5f59e90600ddp1, 0x1.41bfee2424771p2,
0x1.44e6c595afdccp3, -0x1.44e6c595afdccp3, -0x1.41bfee2424771p2,
-0x1.a5f59e90600ddp1, -0x1.3504f333f9de6p1, -0x1.def13b73c1406p0,
-0x1.7f218e25a7461p0, -0x1.37efd8d87607ep0, -0x1.0000000000000p0,
-0x1.a43002ae4285p-1, -0x1.561b82ab7f99p-1, -0x1.11ab7190834ecp-1,
-0x1.a827999fcef32p-2, -0x1.36a08355c63dcp-2, -0x1.975f5e0553158p-3,
-0x1.936bb8c5b2da2p-4, 0.0000000000000000,
};
// Exceptional cases for tanf.
static constexpr size_t N_EXCEPTS = 6;
constexpr size_t N_EXCEPTS = 6;
static constexpr fputil::ExceptValues<float, N_EXCEPTS> TANF_EXCEPTS{{
constexpr fputil::ExceptValues<float, N_EXCEPTS> TANF_EXCEPTS{{
// (inputs, RZ output, RU offset, RD offset, RN offset)
// x = 0x1.3ae898p39, tan(x) = 0x1.23d43cp12 (RZ)
{0x531d744c, 0x4591ea1e, 1, 0, 1},
// x = 0x1.ada6aap27, tan(x) = 0x1.e80304p-3 (RZ)
{0x4d56d355, 0x3e740182, 1, 0, 0},
// x = 0x1.862064p33, tan(x) = -0x1.8dee56p-3 (RZ)
{0x50431032, 0xbe46f72b, 0, 1, 1},
// x = 0x1.af61dap48, tan(x) = 0x1.60d1c6p-2 (RZ)
{0x57d7b0ed, 0x3eb068e3, 1, 0, 1},
// x = 0x1.dd0d2ap76, tan(x) = -0x1.465f1cp22 (RZ)
{0x65ee8695, 0xcaa32f8e, 0, 1, 0},
// x = 0x1.31fc9ep80, tan(x) = 0x1.3c13eep13 (RZ)
{0x6798fe4f, 0x461e09f7, 1, 0, 0},
// x = 0x1.0088bcp52, tan(x) = 0x1.ca1edp0 (RZ)
{0x5980445e, 0x3fe50f68, 1, 0, 0},
// x = 0x1.f90dfcp72, tan(x) = 0x1.597f9cp-1 (RZ)
{0x63fc86fe, 0x3f2cbfce, 1, 0, 0},
// x = 0x1.a6ce12p86, tan(x) = -0x1.c5612ep-1 (RZ)
{0x6ad36709, 0xbf62b097, 0, 1, 0},
// x = 0x1.6a0b76p102, tan(x) = -0x1.e42a1ep0 (RZ)
{0x72b505bb, 0xbff2150f, 0, 1, 0},
}};
LLVM_LIBC_FUNCTION(float, tanf, (float x)) {
using FPBits = typename fputil::FPBits<float>;
FPBits xbits(x);
constexpr double SIGN[2] = {1.0, -1.0};
double x_sign = SIGN[xbits.uintval() >> 31];
xbits.set_sign(false);
uint32_t x_abs = xbits.uintval();
// Range reduction:
//
// Since tan(x) is an odd function,
// tan(x) = -tan(-x),
// By replacing x with -x if x is negative, we can assume in the following
// that x is non-negative.
//
// We perform a range reduction mod pi/32, so that we ca have a good
// polynomial approximation of tan(x) around [-pi/32, pi/32]. Since tan(x) is
// periodic with period pi, in the first step of range reduction, we find k
// and y such that:
// x = (k + y) * pi/32,
// where k is an integer, and |y| <= 0.5.
// Moreover, we only care about the lowest 5 bits of k, since
// tan((k + 32) * pi/32) = tan(k * pi/32 + pi) = tan(k * pi/32).
// So after the reduction k = k & 31, we can assume that 0 <= k <= 31.
//
// For the second step, since tan(x) has a singularity at pi/2, we need a
// further reduction so that:
// k * pi/32 < pi/2, or equivalently, 0 <= k < 16.
// So if k >= 16, we perform the following transformation:
// tan(x) = tan(x - pi) = tan((k + y) * pi/32 - pi)
// = tan((k - 31 + y - 1) * pi/32)
// = tan((k - 31) * pi/32 + (y - 1) * pi/32)
// = tan(k' * pi/32 + y' * pi/32)
// Notice that we only subtract k by 31, not 32, to make sure that |k'| < 16.
// In fact, the range of k' is: -15 <= k' <= 0.
// But the range of y' is now: -1.5 <= y' <= -0.5.
// If we perform round to zero in the first step of finding k and y, so that
// 0 <= y <= 1, then the range of y' would be -1 <= y' <= 0, then we can
// reduce the degree of polynomial approximation using to approximate
// tan(y* pi/32) by 1 or 2 terms.
// In any case, for simplicity and to reuse the same range reduction as sinf
// and cosf, we opted to use the former range: [-1.5, 1.5] * pi/32 for
// the polynomial approximation step.
//
// Once k and y are computed, we then deduce the answer by the tangent of sum
// formula:
// tan(x) = tan((k + y)*pi/32)
// = (tan(y*pi/32) + tan(k*pi/32)) / (1 - tan(y*pi/32)*tan(k*pi/32))
// The values of tan(k*pi/32) for k = -15..15 are precomputed and stored using
// a vector of 31 doubles. Tan(y*pi/32) is computed using degree-9 minimax
// polynomials generated by Sollya.
bool x_sign = xbits.uintval() >> 31;
uint32_t x_abs = xbits.uintval() & 0x7fff'ffffU;
// |x| < pi/32
if (unlikely(x_abs <= 0x3dc9'0fdbU)) {
@ -173,56 +91,45 @@ LLVM_LIBC_FUNCTION(float, tanf, (float x)) {
return xd * result;
}
// Inf or NaN
if (unlikely(x_abs >= 0x7f80'0000U)) {
if (x_abs == 0x7f80'0000U) {
errno = EDOM;
fputil::set_except(FE_INVALID);
}
return x +
FPBits::build_nan(1 << (fputil::MantissaWidth<float>::VALUE - 1));
// Check for exceptional values
if (unlikely(x_abs == 0x3f8a1f62U)) {
// |x| = 0x1.143ec4p0
float sign = x_sign ? -1.0f : 1.0f;
return fputil::multiply_add(sign, 0x1.ddf9f4p0f, sign * 0x1.1p-24f);
}
int64_t k;
double y;
double xd = static_cast<double>(xbits.get_val());
// Perform the first step of range reduction: find k and y such that
// x = (k + y) * pi/32,
// where k is an integer, and |y| <= 0.5.
if (likely(x_abs < FAST_PASS_BOUND)) {
k = small_range_reduction(xd, y);
} else {
if (auto r = TANF_EXCEPTS.lookup_odd(x_abs, x_sign <= 0.0);
// |x| > 0x1.ada6a8p+27f
if (unlikely(x_abs > 0x4d56'd354U)) {
// Inf or NaN
if (unlikely(x_abs >= 0x7f80'0000U)) {
if (x_abs == 0x7f80'0000U) {
errno = EDOM;
fputil::set_except(FE_INVALID);
}
return x +
FPBits::build_nan(1 << (fputil::MantissaWidth<float>::VALUE - 1));
}
// Other large exceptional values
if (auto r = TANF_EXCEPTS.lookup_odd(x_abs, x_sign);
unlikely(r.has_value()))
return r.value();
fputil::FPBits<float> x_bits(x_abs);
k = large_range_reduction(xd, x_bits.get_exponent(), y);
}
// Only care about the lowest 5 bits of k.
k &= 31;
// Adjust y if k >= 16.
constexpr double ADJUSTMENT[2] = {0.0, -1.0};
y += ADJUSTMENT[k >> 4];
// For |x| >= pi/32, we use the definition of tan(x) function:
// tan(x) = sin(x) / cos(x)
// The we follow the same computations of sin(x) and cos(x) as sinf, cosf,
// and sincosf.
double tan_k = TAN_K_PI_OVER_32[k];
double xd = static_cast<double>(x);
double sin_k, cos_k, sin_y, cosm1_y;
// Degree-10 minimax odd polynomial for tan(y * pi/32)/y generated by Sollya
// with:
// > P = fpminimax(tan(y*pi/32)/y, [|0, 2, 4, 6, 8, 10|], [|D...|], [0, 1.5]);
double ysq = y * y;
double tan_y =
y * fputil::polyeval(ysq, 0x1.921fb54442d17p-4, 0x1.4abbce625e84cp-12,
0x1.466bc669afd51p-20, 0x1.460013a5aae3p-28,
0x1.45de3dc438976p-36, 0x1.4eaeead85bef4p-44);
// Combine the results with the tangent of sum formula:
// tan(x) = tan((k + y)*pi/32)
// = (tan(k*pi/32) + tan(k*pi/32)) / (1 - tan(y*pi/32)*tan(k*pi/32))
return x_sign * (tan_y + tan_k) / fputil::multiply_add(tan_y, -tan_k, 1.0);
sincosf_eval(xd, x_abs, sin_k, cos_k, sin_y, cosm1_y);
// tan(x) = sin(x) / cos(x)
// = (sin_y * cos_k + cos_y * sin_k) / (cos_y * cos_k - sin_y * sin_k)
using fputil::multiply_add;
return multiply_add(sin_y, cos_k, multiply_add(cosm1_y, sin_k, sin_k)) /
multiply_add(sin_y, -sin_k, multiply_add(cosm1_y, cos_k, cos_k));
}
} // namespace __llvm_libc

8
libc/test/src/math/tanf_test.cpp
View File

@ -56,13 +56,14 @@ TEST(LlvmLibcTanfTest, InFloatRange) {
}
TEST(LlvmLibcTanfTest, SpecificBitPatterns) {
constexpr int N = 48;
constexpr int N = 54;
constexpr uint32_t INPUTS[N] = {
0x3a7a'8d2fU, // x = 0x1.f51a5ep-11f
0x3f06'0a92U, // x = pi/6
0x3f3a'dc51U, // x = 0x1.75b8a2p-1f
0x3f49'0fdbU, // x = pi/4
0x3f86'0a92U, // x = pi/3
0x3f8a'1f62U, // x = 0x1.143ec4p+0f
0x3fa7'832aU, // x = 0x1.4f0654p+0f
0x3fc9'0fdbU, // x = pi/2
0x4017'1973U, // x = 0x1.2e32e6p+1f
@ -75,14 +76,18 @@ TEST(LlvmLibcTanfTest, SpecificBitPatterns) {
0x474d'246fU, // x = 0x1.9a48dep+15f
0x4afd'ece4U, // x = 0x1.fbd9c8p+22f
0x4c23'32e9U, // x = 0x1.4665d2p+25f
0x4d56'd355U, // x = 0x1.ada6aap+27f
0x5043'1032U, // x = 0x1.862064p+33f
0x50a3'e87fU, // x = 0x1.47d0fep+34f
0x5239'47f6U, // x = 0x1.728fecp+37f
0x531d'744cU, // x = 0x1.3ae898p+39f
0x53b1'46a6U, // x = 0x1.628d4cp+40f
0x5532'5019U, // x = 0x1.64a032p+43f
0x55ca'fb2aU, // x = 0x1.95f654p+44f
0x57d7'b0edU, // x = 0x1.af61dap+48f
0x588e'f060U, // x = 0x1.1de0cp+50f
0x5922'aa80U, // x = 0x1.4555p+51f
0x5980'445eU, // x = 0x1.0088bcp+52f
0x5aa4'542cU, // x = 0x1.48a858p+54f
0x5c07'bcd0U, // x = 0x1.0f79ap+57f
0x5ebc'fddeU, // x = 0x1.79fbbcp+62f
@ -91,6 +96,7 @@ TEST(LlvmLibcTanfTest, SpecificBitPatterns) {
0x6115'cb11U, // x = 0x1.2b9622p+67f
0x61a4'0b40U, // x = 0x1.48168p+68f
0x6386'134eU, // x = 0x1.0c269cp+72f
0x63fc'86feU, // x = 0x1.f90dfcp+72f
0x6589'8498U, // x = 0x1.13093p+76f
0x65ee'8695U, // x = 0x1.dd0d2ap+76f
0x6600'0001U, // x = 0x1.000002p+77f