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[libc][math] Implement fast pass for double precision atan function. (#132333)

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Implement fast pass for double precision `atan` using range reduction
modulo 1/64 and degree-9 Taylor polynomial.
Relative error is bounded by 2^-66.
This commit is contained in:
lntue 2025-03-21 14:12:06 -04:00 committed by GitHub
parent 8d78b7cc7d
commit a17b03f0e4
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GPG Key ID: B5690EEEBB952194
19 changed files with 512 additions and 124 deletions
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1
libc/config/darwin/arm/entrypoints.txt
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@ -126,6 +126,7 @@ set(TARGET_LIBM_ENTRYPOINTS
libc.src.math.asinhf
libc.src.math.atan2
libc.src.math.atan2f
libc.src.math.atan
libc.src.math.atanf
libc.src.math.atanhf
libc.src.math.cbrt

1
libc/config/linux/aarch64/entrypoints.txt
View File

@ -411,6 +411,7 @@ set(TARGET_LIBM_ENTRYPOINTS
libc.src.math.asinhf
libc.src.math.atan2
libc.src.math.atan2f
libc.src.math.atan
libc.src.math.atanf
libc.src.math.atanhf
libc.src.math.canonicalize

1
libc/config/linux/arm/entrypoints.txt
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@ -244,6 +244,7 @@ set(TARGET_LIBM_ENTRYPOINTS
libc.src.math.asinhf
libc.src.math.atan2
libc.src.math.atan2f
libc.src.math.atan
libc.src.math.atanf
libc.src.math.atanhf
libc.src.math.cbrt

1
libc/config/linux/riscv/entrypoints.txt
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@ -401,6 +401,7 @@ set(TARGET_LIBM_ENTRYPOINTS
libc.src.math.asinhf
libc.src.math.atan2
libc.src.math.atan2f
libc.src.math.atan
libc.src.math.atanf
libc.src.math.atanhf
libc.src.math.canonicalize

1
libc/config/linux/x86_64/entrypoints.txt
View File

@ -413,6 +413,7 @@ set(TARGET_LIBM_ENTRYPOINTS
libc.src.math.asinhf
libc.src.math.atan2
libc.src.math.atan2f
libc.src.math.atan
libc.src.math.atanf
libc.src.math.atanhf
libc.src.math.canonicalize

1
libc/config/windows/entrypoints.txt
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@ -132,6 +132,7 @@ set(TARGET_LIBM_ENTRYPOINTS
libc.src.math.asinhf
libc.src.math.atan2
libc.src.math.atan2f
libc.src.math.atan
libc.src.math.atanf
libc.src.math.atanhf
libc.src.math.cbrt

2
libc/docs/headers/math/index.rst
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@ -261,7 +261,7 @@ Higher Math Functions
+-----------+------------------+-----------------+------------------------+----------------------+------------------------+------------------------+----------------------------+
| asinpi | | | | | | 7.12.4.9 | F.10.1.9 |
+-----------+------------------+-----------------+------------------------+----------------------+------------------------+------------------------+----------------------------+
| atan | |check| | | | | | 7.12.4.3 | F.10.1.3 |
| atan | |check| | 1 ULP | | | | 7.12.4.3 | F.10.1.3 |
+-----------+------------------+-----------------+------------------------+----------------------+------------------------+------------------------+----------------------------+
| atan2 | |check| | 1 ULP | | | | 7.12.4.4 | F.10.1.4 |
+-----------+------------------+-----------------+------------------------+----------------------+------------------------+------------------------+----------------------------+

6
libc/include/math.yaml
View File

@ -52,6 +52,12 @@ functions:
return_type: float
arguments:
- type: float
- name: atan
standards:
- stdc
return_type: double
arguments:
- type: double
- name: atan2
standards:
- stdc

33
libc/src/math/generic/CMakeLists.txt
View File

@ -4054,6 +4054,16 @@ add_entrypoint_object(
libc.src.__support.macros.properties.types
)
add_header_library(
atan_utils
HDRS
atan_utils.h
DEPENDS
libc.src.__support.FPUtil.double_double
libc.src.__support.FPUtil.multiply_add
libc.src.__support.macros.optimization
)
add_entrypoint_object(
atanf
SRCS
@ -4071,6 +4081,24 @@ add_entrypoint_object(
libc.src.__support.macros.optimization
)
add_entrypoint_object(
atan
SRCS
atan.cpp
HDRS
../atan.h
COMPILE_OPTIONS
-O3
DEPENDS
.atan_utils
libc.src.__support.FPUtil.double_double
libc.src.__support.FPUtil.fenv_impl
libc.src.__support.FPUtil.fp_bits
libc.src.__support.FPUtil.multiply_add
libc.src.__support.FPUtil.nearest_integer
libc.src.__support.macros.optimization
)
add_entrypoint_object(
atan2f
SRCS
@ -4096,14 +4124,11 @@ add_entrypoint_object(
HDRS
../atan2.h
DEPENDS
.inv_trigf_utils
.atan_utils
libc.src.__support.FPUtil.double_double
libc.src.__support.FPUtil.dyadic_float
libc.src.__support.FPUtil.fp_bits
libc.src.__support.FPUtil.multiply_add
libc.src.__support.FPUtil.nearest_integer
libc.src.__support.FPUtil.polyeval
libc.src.__support.FPUtil.rounding_mode
libc.src.__support.macros.optimization
)

179
libc/src/math/generic/atan.cpp Normal file
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@ -0,0 +1,179 @@
//===-- Double-precision atan function ------------------------------------===//
//
// Part of the LLVM Project, under the Apache License v2.0 with LLVM Exceptions.
// See https://llvm.org/LICENSE.txt for license information.
// SPDX-License-Identifier: Apache-2.0 WITH LLVM-exception
//
//===----------------------------------------------------------------------===//
#include "src/math/atan.h"
#include "atan_utils.h"
#include "src/__support/FPUtil/FEnvImpl.h"
#include "src/__support/FPUtil/FPBits.h"
#include "src/__support/FPUtil/double_double.h"
#include "src/__support/FPUtil/multiply_add.h"
#include "src/__support/FPUtil/nearest_integer.h"
#include "src/__support/macros/config.h"
#include "src/__support/macros/optimization.h" // LIBC_UNLIKELY
namespace LIBC_NAMESPACE_DECL {
// To compute atan(x), we divided it into the following cases:
// * |x| < 2^-26:
// Since |x| > atan(|x|) > |x| - |x|^3/3, and |x|^3/3 < ulp(x)/2, we simply
// return atan(x) = x - sign(x) * epsilon.
// * 2^-26 <= |x| < 1:
// We perform range reduction mod 2^-6 = 1/64 as follow:
// Let k = 2^(-6) * round(|x| * 2^6), then
// atan(x) = sign(x) * atan(|x|)
// = sign(x) * (atan(k) + atan((|x| - k) / (1 + |x|*k)).
// We store atan(k) in a look up table, and perform intermediate steps in
// double-double.
// * 1 < |x| < 2^53:
// First we perform the transformation y = 1/|x|:
// atan(x) = sign(x) * (pi/2 - atan(1/|x|))
// = sign(x) * (pi/2 - atan(y)).
// Then we compute atan(y) using range reduction mod 2^-6 = 1/64 as the
// previous case:
// Let k = 2^(-6) * round(y * 2^6), then
// atan(y) = atan(k) + atan((y - k) / (1 + y*k))
// = atan(k) + atan((1/|x| - k) / (1 + k/|x|)
// = atan(k) + atan((1 - k*|x|) / (|x| + k)).
// * |x| >= 2^53:
// Using the reciprocal transformation:
// atan(x) = sign(x) * (pi/2 - atan(1/|x|)).
// We have that:
// atan(1/|x|) <= 1/|x| <= 2^-53,
// which is smaller than ulp(pi/2) / 2.
// So we can return:
// atan(x) = sign(x) * (pi/2 - epsilon)
LLVM_LIBC_FUNCTION(double, atan, (double x)) {
using FPBits = fputil::FPBits<double>;
constexpr double IS_NEG[2] = {1.0, -1.0};
constexpr DoubleDouble PI_OVER_2 = {0x1.1a62633145c07p-54,
0x1.921fb54442d18p0};
constexpr DoubleDouble MPI_OVER_2 = {-0x1.1a62633145c07p-54,
-0x1.921fb54442d18p0};
FPBits xbits(x);
bool x_sign = xbits.is_neg();
xbits = xbits.abs();
uint64_t x_abs = xbits.uintval();
int x_exp =
static_cast<int>(x_abs >> FPBits::FRACTION_LEN) - FPBits::EXP_BIAS;
// |x| < 1.
if (x_exp < 0) {
if (LIBC_UNLIKELY(x_exp < -26)) {
#ifdef LIBC_MATH_HAS_SKIP_ACCURATE_PASS
return x;
#else
if (x == 0.0)
return x;
// |x| < 2^-26
return fputil::multiply_add(-0x1.0p-54, x, x);
#endif // LIBC_MATH_HAS_SKIP_ACCURATE_PASS
}
double x_d = xbits.get_val();
// k = 2^-6 * round(2^6 * |x|)
double k = fputil::nearest_integer(0x1.0p6 * x_d);
unsigned idx = static_cast<unsigned>(k);
k *= 0x1.0p-6;
// numerator = |x| - k
DoubleDouble num, den;
num.lo = 0.0;
num.hi = x_d - k;
// denominator = 1 - k * |x|
den.hi = fputil::multiply_add(x_d, k, 1.0);
DoubleDouble prod = fputil::exact_mult(x_d, k);
// Using Dekker's 2SUM algorithm to compute the lower part.
den.lo = ((1.0 - den.hi) + prod.hi) + prod.lo;
// x_r = (|x| - k) / (1 + k * |x|)
DoubleDouble x_r = fputil::div(num, den);
// Approximating atan(x_r) using Taylor polynomial.
DoubleDouble p = atan_eval(x_r);
// atan(x) = sign(x) * (atan(k) + atan(x_r))
// = sign(x) * (atan(k) + atan( (|x| - k) / (1 + k * |x|) ))
#ifdef LIBC_MATH_HAS_SKIP_ACCURATE_PASS
return IS_NEG[x_sign] * (ATAN_I[idx].hi + (p.hi + (p.lo + ATAN_I[idx].lo)));
#else
DoubleDouble c0 = fputil::exact_add(ATAN_I[idx].hi, p.hi);
double c1 = c0.lo + (ATAN_I[idx].lo + p.lo);
double r = IS_NEG[x_sign] * (c0.hi + c1);
return r;
#endif // LIBC_MATH_HAS_SKIP_ACCURATE_PASS
}
// |x| >= 2^53 or x is NaN.
if (LIBC_UNLIKELY(x_exp >= 53)) {
// x is nan
if (xbits.is_nan()) {
if (xbits.is_signaling_nan()) {
fputil::raise_except_if_required(FE_INVALID);
return FPBits::quiet_nan().get_val();
}
return x;
}
// |x| >= 2^53
// atan(x) ~ sign(x) * pi/2.
if (x_exp >= 53)
#ifdef LIBC_MATH_HAS_SKIP_ACCURATE_PASS
return IS_NEG[x_sign] * PI_OVER_2.hi;
#else
return fputil::multiply_add(IS_NEG[x_sign], PI_OVER_2.hi,
IS_NEG[x_sign] * PI_OVER_2.lo);
#endif // LIBC_MATH_HAS_SKIP_ACCURATE_PASS
}
double x_d = xbits.get_val();
double y = 1.0 / x_d;
// k = 2^-6 * round(2^6 / |x|)
double k = fputil::nearest_integer(0x1.0p6 * y);
unsigned idx = static_cast<unsigned>(k);
k *= 0x1.0p-6;
// denominator = |x| + k
DoubleDouble den = fputil::exact_add(x_d, k);
// numerator = 1 - k * |x|
DoubleDouble num;
num.hi = fputil::multiply_add(-x_d, k, 1.0);
DoubleDouble prod = fputil::exact_mult(x_d, k);
// Using Dekker's 2SUM algorithm to compute the lower part.
num.lo = ((1.0 - num.hi) - prod.hi) - prod.lo;
// x_r = (1/|x| - k) / (1 - k/|x|)
// = (1 - k * |x|) / (|x| - k)
DoubleDouble x_r = fputil::div(num, den);
// Approximating atan(x_r) using Taylor polynomial.
DoubleDouble p = atan_eval(x_r);
// atan(x) = sign(x) * (pi/2 - atan(1/|x|))
// = sign(x) * (pi/2 - atan(k) - atan(x_r))
// = (-sign(x)) * (-pi/2 + atan(k) + atan((1 - k*|x|)/(|x| - k)))
#ifdef LIBC_MATH_HAS_SKIP_ACCURATE_PASS
double lo_part = p.lo + ATAN_I[idx].lo + MPI_OVER_2.lo;
return IS_NEG[!x_sign] * (MPI_OVER_2.hi + ATAN_I[idx].hi + (p.hi + lo_part));
#else
DoubleDouble c0 = fputil::exact_add(MPI_OVER_2.hi, ATAN_I[idx].hi);
DoubleDouble c1 = fputil::exact_add(c0.hi, p.hi);
double c2 = c1.lo + (c0.lo + p.lo) + (ATAN_I[idx].lo + MPI_OVER_2.lo);
double r = IS_NEG[!x_sign] * (c1.hi + c2);
return r;
#endif
}
} // namespace LIBC_NAMESPACE_DECL

119
libc/src/math/generic/atan2.cpp
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@ -7,133 +7,16 @@
//===----------------------------------------------------------------------===//
#include "src/math/atan2.h"
#include "inv_trigf_utils.h"
#include "atan_utils.h"
#include "src/__support/FPUtil/FPBits.h"
#include "src/__support/FPUtil/PolyEval.h"
#include "src/__support/FPUtil/double_double.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/macros/config.h"
#include "src/__support/macros/optimization.h" // LIBC_UNLIKELY
namespace LIBC_NAMESPACE_DECL {
namespace {
using DoubleDouble = fputil::DoubleDouble;
// atan(i/64) with i = 0..64, generated by Sollya with:
// > for i from 0 to 64 do {
// a = round(atan(i/64), D, RN);
// b = round(atan(i/64) - a, D, RN);
// print("{", b, ",", a, "},");
// };
constexpr fputil::DoubleDouble ATAN_I[65] = {
{0.0, 0.0},
{-0x1.220c39d4dff5p-61, 0x1.fff555bbb729bp-7},
{-0x1.5ec431444912cp-60, 0x1.ffd55bba97625p-6},
{-0x1.86ef8f794f105p-63, 0x1.7fb818430da2ap-5},
{-0x1.c934d86d23f1dp-60, 0x1.ff55bb72cfdeap-5},
{0x1.ac4ce285df847p-58, 0x1.3f59f0e7c559dp-4},
{-0x1.cfb654c0c3d98p-58, 0x1.7ee182602f10fp-4},
{0x1.f7b8f29a05987p-58, 0x1.be39ebe6f07c3p-4},
{-0x1.cd37686760c17p-59, 0x1.fd5ba9aac2f6ep-4},
{-0x1.b485914dacf8cp-59, 0x1.1e1fafb043727p-3},
{0x1.61a3b0ce9281bp-57, 0x1.3d6eee8c6626cp-3},
{-0x1.054ab2c010f3dp-58, 0x1.5c9811e3ec26ap-3},
{0x1.347b0b4f881cap-58, 0x1.7b97b4bce5b02p-3},
{0x1.cf601e7b4348ep-59, 0x1.9a6a8e96c8626p-3},
{0x1.17b10d2e0e5abp-61, 0x1.b90d7529260a2p-3},
{0x1.c648d1534597ep-57, 0x1.d77d5df205736p-3},
{0x1.8ab6e3cf7afbdp-57, 0x1.f5b75f92c80ddp-3},
{0x1.62e47390cb865p-56, 0x1.09dc597d86362p-2},
{0x1.30ca4748b1bf9p-57, 0x1.18bf5a30bf178p-2},
{-0x1.077cdd36dfc81p-56, 0x1.278372057ef46p-2},
{-0x1.963a544b672d8p-57, 0x1.362773707ebccp-2},
{-0x1.5d5e43c55b3bap-56, 0x1.44aa436c2af0ap-2},
{-0x1.2566480884082p-57, 0x1.530ad9951cd4ap-2},
{-0x1.a725715711fp-56, 0x1.614840309cfe2p-2},
{-0x1.c63aae6f6e918p-56, 0x1.6f61941e4def1p-2},
{0x1.69c885c2b249ap-56, 0x1.7d5604b63b3f7p-2},
{0x1.b6d0ba3748fa8p-56, 0x1.8b24d394a1b25p-2},
{0x1.9e6c988fd0a77p-56, 0x1.98cd5454d6b18p-2},
{-0x1.24dec1b50b7ffp-56, 0x1.a64eec3cc23fdp-2},
{0x1.ae187b1ca504p-56, 0x1.b3a911da65c6cp-2},
{-0x1.cc1ce70934c34p-56, 0x1.c0db4c94ec9fp-2},
{-0x1.a2cfa4418f1adp-56, 0x1.cde53432c1351p-2},
{0x1.a2b7f222f65e2p-56, 0x1.dac670561bb4fp-2},
{0x1.0e53dc1bf3435p-56, 0x1.e77eb7f175a34p-2},
{-0x1.a3992dc382a23p-57, 0x1.f40dd0b541418p-2},
{-0x1.b32c949c9d593p-55, 0x1.0039c73c1a40cp-1},
{-0x1.d5b495f6349e6p-56, 0x1.0657e94db30dp-1},
{0x1.974fa13b5404fp-58, 0x1.0c6145b5b43dap-1},
{-0x1.2bdaee1c0ee35p-58, 0x1.1255d9bfbd2a9p-1},
{0x1.c621cec00c301p-55, 0x1.1835a88be7c13p-1},
{-0x1.928df287a668fp-58, 0x1.1e00babdefeb4p-1},
{0x1.c421c9f38224ep-57, 0x1.23b71e2cc9e6ap-1},
{-0x1.09e73b0c6c087p-56, 0x1.2958e59308e31p-1},
{0x1.c5d5e9ff0cf8dp-55, 0x1.2ee628406cbcap-1},
{0x1.1021137c71102p-55, 0x1.345f01cce37bbp-1},
{-0x1.2304331d8bf46p-55, 0x1.39c391cd4171ap-1},
{0x1.ecf8b492644fp-56, 0x1.3f13fb89e96f4p-1},
{-0x1.f76d0163f79c8p-56, 0x1.445065b795b56p-1},
{0x1.2419a87f2a458p-56, 0x1.4978fa3269ee1p-1},
{0x1.4a33dbeb3796cp-55, 0x1.4e8de5bb6ec04p-1},
{-0x1.1bb74abda520cp-55, 0x1.538f57b89061fp-1},
{-0x1.5e5c9d8c5a95p-56, 0x1.587d81f732fbbp-1},
{0x1.0028e4bc5e7cap-57, 0x1.5d58987169b18p-1},
{-0x1.2b785350ee8c1p-57, 0x1.6220d115d7b8ep-1},
{-0x1.6ea6febe8bbbap-56, 0x1.66d663923e087p-1},
{-0x1.a80386188c50ep-55, 0x1.6b798920b3d99p-1},
{-0x1.8c34d25aadef6p-56, 0x1.700a7c5784634p-1},
{0x1.7b2a6165884a1p-59, 0x1.748978fba8e0fp-1},
{0x1.406a08980374p-55, 0x1.78f6bbd5d315ep-1},
{0x1.560821e2f3aa9p-55, 0x1.7d528289fa093p-1},
{-0x1.bf76229d3b917p-56, 0x1.819d0b7158a4dp-1},
{0x1.6b66e7fc8b8c3p-57, 0x1.85d69576cc2c5p-1},
{-0x1.55b9a5e177a1bp-55, 0x1.89ff5ff57f1f8p-1},
{-0x1.ec182ab042f61p-56, 0x1.8e17aa99cc05ep-1},
{0x1.1a62633145c07p-55, 0x1.921fb54442d18p-1},
};
// Approximate atan(x) for |x| <= 2^-7.
// Using degree-9 Taylor polynomial:
// P = x - x^3/3 + x^5/5 -x^7/7 + x^9/9;
// Then the absolute error is bounded by:
// |atan(x) - P(x)| < |x|^11/11 < 2^(-7*11) / 11 < 2^-80.
// And the relative error is bounded by:
// |(atan(x) - P(x))/atan(x)| < |x|^10 / 10 < 2^-73.
// For x = x_hi + x_lo, fully expand the polynomial and drop any terms less than
// ulp(x_hi^3 / 3) gives us:
// P(x) ~ x_hi - x_hi^3/3 + x_hi^5/5 - x_hi^7/7 + x_hi^9/9 +
// + x_lo * (1 - x_hi^2 + x_hi^4)
DoubleDouble atan_eval(const DoubleDouble &x) {
DoubleDouble p;
p.hi = x.hi;
double x_hi_sq = x.hi * x.hi;
// c0 ~ x_hi^2 * 1/5 - 1/3
double c0 = fputil::multiply_add(x_hi_sq, 0x1.999999999999ap-3,
-0x1.5555555555555p-2);
// c1 ~ x_hi^2 * 1/9 - 1/7
double c1 = fputil::multiply_add(x_hi_sq, 0x1.c71c71c71c71cp-4,
-0x1.2492492492492p-3);
// x_hi^3
double x_hi_3 = x_hi_sq * x.hi;
// x_hi^4
double x_hi_4 = x_hi_sq * x_hi_sq;
// d0 ~ 1/3 - x_hi^2 / 5 + x_hi^4 / 7 - x_hi^6 / 9
double d0 = fputil::multiply_add(x_hi_4, c1, c0);
// x_lo - x_lo * x_hi^2 + x_lo * x_hi^4
double d1 = fputil::multiply_add(x_hi_4 - x_hi_sq, x.lo, x.lo);
// p.lo ~ -x_hi^3/3 + x_hi^5/5 - x_hi^7/7 + x_hi^9/9 +
// + x_lo * (1 - x_hi^2 + x_hi^4)
p.lo = fputil::multiply_add(x_hi_3, d0, d1);
return p;
}
} // anonymous namespace
// There are several range reduction steps we can take for atan2(y, x) as
// follow:

137
libc/src/math/generic/atan_utils.h Normal file
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@ -0,0 +1,137 @@
//===-- Collection of utils for atan/atan2 ----------------------*- C++ -*-===//
//
// Part of the LLVM Project, under the Apache License v2.0 with LLVM Exceptions.
// See https://llvm.org/LICENSE.txt for license information.
// SPDX-License-Identifier: Apache-2.0 WITH LLVM-exception
//
//===----------------------------------------------------------------------===//
#ifndef LLVM_LIBC_SRC_MATH_GENERIC_ATAN_UTILS_H
#define LLVM_LIBC_SRC_MATH_GENERIC_ATAN_UTILS_H
#include "src/__support/FPUtil/double_double.h"
#include "src/__support/FPUtil/multiply_add.h"
#include "src/__support/macros/config.h"
namespace LIBC_NAMESPACE_DECL {
namespace {
using DoubleDouble = fputil::DoubleDouble;
// atan(i/64) with i = 0..64, generated by Sollya with:
// > for i from 0 to 64 do {
// a = round(atan(i/64), D, RN);
// b = round(atan(i/64) - a, D, RN);
// print("{", b, ",", a, "},");
// };
constexpr fputil::DoubleDouble ATAN_I[65] = {
{0.0, 0.0},
{-0x1.220c39d4dff5p-61, 0x1.fff555bbb729bp-7},
{-0x1.5ec431444912cp-60, 0x1.ffd55bba97625p-6},
{-0x1.86ef8f794f105p-63, 0x1.7fb818430da2ap-5},
{-0x1.c934d86d23f1dp-60, 0x1.ff55bb72cfdeap-5},
{0x1.ac4ce285df847p-58, 0x1.3f59f0e7c559dp-4},
{-0x1.cfb654c0c3d98p-58, 0x1.7ee182602f10fp-4},
{0x1.f7b8f29a05987p-58, 0x1.be39ebe6f07c3p-4},
{-0x1.cd37686760c17p-59, 0x1.fd5ba9aac2f6ep-4},
{-0x1.b485914dacf8cp-59, 0x1.1e1fafb043727p-3},
{0x1.61a3b0ce9281bp-57, 0x1.3d6eee8c6626cp-3},
{-0x1.054ab2c010f3dp-58, 0x1.5c9811e3ec26ap-3},
{0x1.347b0b4f881cap-58, 0x1.7b97b4bce5b02p-3},
{0x1.cf601e7b4348ep-59, 0x1.9a6a8e96c8626p-3},
{0x1.17b10d2e0e5abp-61, 0x1.b90d7529260a2p-3},
{0x1.c648d1534597ep-57, 0x1.d77d5df205736p-3},
{0x1.8ab6e3cf7afbdp-57, 0x1.f5b75f92c80ddp-3},
{0x1.62e47390cb865p-56, 0x1.09dc597d86362p-2},
{0x1.30ca4748b1bf9p-57, 0x1.18bf5a30bf178p-2},
{-0x1.077cdd36dfc81p-56, 0x1.278372057ef46p-2},
{-0x1.963a544b672d8p-57, 0x1.362773707ebccp-2},
{-0x1.5d5e43c55b3bap-56, 0x1.44aa436c2af0ap-2},
{-0x1.2566480884082p-57, 0x1.530ad9951cd4ap-2},
{-0x1.a725715711fp-56, 0x1.614840309cfe2p-2},
{-0x1.c63aae6f6e918p-56, 0x1.6f61941e4def1p-2},
{0x1.69c885c2b249ap-56, 0x1.7d5604b63b3f7p-2},
{0x1.b6d0ba3748fa8p-56, 0x1.8b24d394a1b25p-2},
{0x1.9e6c988fd0a77p-56, 0x1.98cd5454d6b18p-2},
{-0x1.24dec1b50b7ffp-56, 0x1.a64eec3cc23fdp-2},
{0x1.ae187b1ca504p-56, 0x1.b3a911da65c6cp-2},
{-0x1.cc1ce70934c34p-56, 0x1.c0db4c94ec9fp-2},
{-0x1.a2cfa4418f1adp-56, 0x1.cde53432c1351p-2},
{0x1.a2b7f222f65e2p-56, 0x1.dac670561bb4fp-2},
{0x1.0e53dc1bf3435p-56, 0x1.e77eb7f175a34p-2},
{-0x1.a3992dc382a23p-57, 0x1.f40dd0b541418p-2},
{-0x1.b32c949c9d593p-55, 0x1.0039c73c1a40cp-1},
{-0x1.d5b495f6349e6p-56, 0x1.0657e94db30dp-1},
{0x1.974fa13b5404fp-58, 0x1.0c6145b5b43dap-1},
{-0x1.2bdaee1c0ee35p-58, 0x1.1255d9bfbd2a9p-1},
{0x1.c621cec00c301p-55, 0x1.1835a88be7c13p-1},
{-0x1.928df287a668fp-58, 0x1.1e00babdefeb4p-1},
{0x1.c421c9f38224ep-57, 0x1.23b71e2cc9e6ap-1},
{-0x1.09e73b0c6c087p-56, 0x1.2958e59308e31p-1},
{0x1.c5d5e9ff0cf8dp-55, 0x1.2ee628406cbcap-1},
{0x1.1021137c71102p-55, 0x1.345f01cce37bbp-1},
{-0x1.2304331d8bf46p-55, 0x1.39c391cd4171ap-1},
{0x1.ecf8b492644fp-56, 0x1.3f13fb89e96f4p-1},
{-0x1.f76d0163f79c8p-56, 0x1.445065b795b56p-1},
{0x1.2419a87f2a458p-56, 0x1.4978fa3269ee1p-1},
{0x1.4a33dbeb3796cp-55, 0x1.4e8de5bb6ec04p-1},
{-0x1.1bb74abda520cp-55, 0x1.538f57b89061fp-1},
{-0x1.5e5c9d8c5a95p-56, 0x1.587d81f732fbbp-1},
{0x1.0028e4bc5e7cap-57, 0x1.5d58987169b18p-1},
{-0x1.2b785350ee8c1p-57, 0x1.6220d115d7b8ep-1},
{-0x1.6ea6febe8bbbap-56, 0x1.66d663923e087p-1},
{-0x1.a80386188c50ep-55, 0x1.6b798920b3d99p-1},
{-0x1.8c34d25aadef6p-56, 0x1.700a7c5784634p-1},
{0x1.7b2a6165884a1p-59, 0x1.748978fba8e0fp-1},
{0x1.406a08980374p-55, 0x1.78f6bbd5d315ep-1},
{0x1.560821e2f3aa9p-55, 0x1.7d528289fa093p-1},
{-0x1.bf76229d3b917p-56, 0x1.819d0b7158a4dp-1},
{0x1.6b66e7fc8b8c3p-57, 0x1.85d69576cc2c5p-1},
{-0x1.55b9a5e177a1bp-55, 0x1.89ff5ff57f1f8p-1},
{-0x1.ec182ab042f61p-56, 0x1.8e17aa99cc05ep-1},
{0x1.1a62633145c07p-55, 0x1.921fb54442d18p-1},
};
// Approximate atan(x) for |x| <= 2^-7.
// Using degree-9 Taylor polynomial:
// P = x - x^3/3 + x^5/5 -x^7/7 + x^9/9;
// Then the absolute error is bounded by:
// |atan(x) - P(x)| < |x|^11/11 < 2^(-7*11) / 11 < 2^-80.
// And the relative error is bounded by:
// |(atan(x) - P(x))/atan(x)| < |x|^10 / 10 < 2^-73.
// For x = x_hi + x_lo, fully expand the polynomial and drop any terms less than
// ulp(x_hi^3 / 3) gives us:
// P(x) ~ x_hi - x_hi^3/3 + x_hi^5/5 - x_hi^7/7 + x_hi^9/9 +
// + x_lo * (1 - x_hi^2 + x_hi^4)
// Since p.lo is ~ x^3/3, the relative error from rounding is bounded by:
// |(atan(x) - P(x))/atan(x)| < ulp(x^2) <= 2^(-14-52) = 2^-66.
DoubleDouble atan_eval(const DoubleDouble &x) {
DoubleDouble p;
p.hi = x.hi;
double x_hi_sq = x.hi * x.hi;
// c0 ~ x_hi^2 * 1/5 - 1/3
double c0 = fputil::multiply_add(x_hi_sq, 0x1.999999999999ap-3,
-0x1.5555555555555p-2);
// c1 ~ x_hi^2 * 1/9 - 1/7
double c1 = fputil::multiply_add(x_hi_sq, 0x1.c71c71c71c71cp-4,
-0x1.2492492492492p-3);
// x_hi^3
double x_hi_3 = x_hi_sq * x.hi;
// x_hi^4
double x_hi_4 = x_hi_sq * x_hi_sq;
// d0 ~ 1/3 - x_hi^2 / 5 + x_hi^4 / 7 - x_hi^6 / 9
double d0 = fputil::multiply_add(x_hi_4, c1, c0);
// x_lo - x_lo * x_hi^2 + x_lo * x_hi^4
double d1 = fputil::multiply_add(x_hi_4 - x_hi_sq, x.lo, x.lo);
// p.lo ~ -x_hi^3/3 + x_hi^5/5 - x_hi^7/7 + x_hi^9/9 +
// + x_lo * (1 - x_hi^2 + x_hi^4)
p.lo = fputil::multiply_add(x_hi_3, d0, d1);
return p;
}
} // anonymous namespace
} // namespace LIBC_NAMESPACE_DECL
#endif // LLVM_LIBC_SRC_MATH_GENERIC_ATAN_UTILS_H

12
libc/test/src/math/CMakeLists.txt
View File

@ -2234,6 +2234,18 @@ add_fp_unittest(
libc.src.__support.FPUtil.fp_bits
)
add_fp_unittest(
atan_test
NEED_MPFR
SUITE
libc-math-unittests
SRCS
atan_test.cpp
DEPENDS
libc.src.math.atan
libc.src.__support.FPUtil.fp_bits
)
add_fp_unittest(
scalbn_test
NEED_MPFR

84
libc/test/src/math/atan_test.cpp Normal file
View File

@ -0,0 +1,84 @@
//===-- Unittests for atan ------------------------------------------------===//
//
// Part of the LLVM Project, under the Apache License v2.0 with LLVM Exceptions.
// See https://llvm.org/LICENSE.txt for license information.
// SPDX-License-Identifier: Apache-2.0 WITH LLVM-exception
//
//===----------------------------------------------------------------------===//
#include "src/__support/FPUtil/FPBits.h"
#include "src/math/atan.h"
#include "test/UnitTest/FPMatcher.h"
#include "test/UnitTest/Test.h"
#include "utils/MPFRWrapper/MPFRUtils.h"
using LlvmLibcAtanTest = LIBC_NAMESPACE::testing::FPTest<double>;
namespace mpfr = LIBC_NAMESPACE::testing::mpfr;
using LIBC_NAMESPACE::testing::tlog;
TEST_F(LlvmLibcAtanTest, InDoubleRange) {
constexpr uint64_t COUNT = 123'451;
uint64_t START = LIBC_NAMESPACE::fputil::FPBits<double>(0x1.0p-60).uintval();
uint64_t STOP = LIBC_NAMESPACE::fputil::FPBits<double>(0x1.0p60).uintval();
uint64_t STEP = (STOP - START) / COUNT;
auto test = [&](mpfr::RoundingMode rounding_mode) {
mpfr::ForceRoundingMode __r(rounding_mode);
if (!__r.success)
return;
uint64_t fails = 0;
uint64_t count = 0;
uint64_t cc = 0;
double mx, mr = 0.0;
double tol = 0.5;
for (uint64_t i = 0, v = START; i <= COUNT; ++i, v += STEP) {
double x = FPBits(v).get_val();
if (FPBits(v).is_nan() || FPBits(v).is_inf())
continue;
LIBC_NAMESPACE::libc_errno = 0;
double result = LIBC_NAMESPACE::atan(x);
++cc;
if (FPBits(result).is_nan() || FPBits(result).is_inf())
continue;
++count;
if (!TEST_MPFR_MATCH_ROUNDING_SILENTLY(mpfr::Operation::Atan, x, result,
2.0, rounding_mode)) {
++fails;
while (!TEST_MPFR_MATCH_ROUNDING_SILENTLY(mpfr::Operation::Atan, x,
result, tol, rounding_mode)) {
mx = x;
mr = result;
if (tol > 1000.0)
break;
tol *= 2.0;
}
}
}
if (fails) {
tlog << " Atan failed: " << fails << "/" << count << "/" << cc
<< " tests.\n";
tlog << " Max ULPs is at most: " << static_cast<uint64_t>(tol) << ".\n";
EXPECT_MPFR_MATCH(mpfr::Operation::Atan, mx, mr, 0.5, rounding_mode);
}
};
tlog << " Test Rounding To Nearest...\n";
test(mpfr::RoundingMode::Nearest);
tlog << " Test Rounding Downward...\n";
test(mpfr::RoundingMode::Downward);
tlog << " Test Rounding Upward...\n";
test(mpfr::RoundingMode::Upward);
tlog << " Test Rounding Toward Zero...\n";
test(mpfr::RoundingMode::TowardZero);
}

10
libc/test/src/math/smoke/CMakeLists.txt
View File

@ -4003,6 +4003,16 @@ add_fp_unittest(
libc.src.__support.FPUtil.fp_bits
)
add_fp_unittest(
atan_test
SUITE
libc-math-smoke-tests
SRCS
atan_test.cpp
DEPENDS
libc.src.math.atan
)
add_fp_unittest(
atan2f_test
SUITE

28
libc/test/src/math/smoke/atan_test.cpp Normal file
View File

@ -0,0 +1,28 @@
//===-- Unittests for atan ------------------------------------------------===//
//
// Part of the LLVM Project, under the Apache License v2.0 with LLVM Exceptions.
// See https://llvm.org/LICENSE.txt for license information.
// SPDX-License-Identifier: Apache-2.0 WITH LLVM-exception
//
//===----------------------------------------------------------------------===//
#include "src/math/atan.h"
#include "test/UnitTest/FPMatcher.h"
#include "test/UnitTest/Test.h"
using LlvmLibcAtanTest = LIBC_NAMESPACE::testing::FPTest<double>;
TEST_F(LlvmLibcAtanTest, SpecialNumbers) {
EXPECT_FP_EQ_ALL_ROUNDING(aNaN, LIBC_NAMESPACE::atan(aNaN));
// atan(sNaN) = aNaN.
EXPECT_EQ(FPBits(aNaN).uintval(),
FPBits(LIBC_NAMESPACE::atan(sNaN)).uintval());
EXPECT_FP_EQ_ALL_ROUNDING(zero, LIBC_NAMESPACE::atan(zero));
EXPECT_FP_EQ_ALL_ROUNDING(neg_zero, LIBC_NAMESPACE::atan(neg_zero));
// atan(+-Inf) = +- pi/2.
EXPECT_FP_EQ(0x1.921fb54442d18p0, LIBC_NAMESPACE::atan(inf));
EXPECT_FP_EQ(-0x1.921fb54442d18p0, LIBC_NAMESPACE::atan(neg_inf));
// atan(+-1) = +- pi/4.
EXPECT_FP_EQ(0x1.921fb54442d18p-1, LIBC_NAMESPACE::atan(1.0));
EXPECT_FP_EQ(-0x1.921fb54442d18p-1, LIBC_NAMESPACE::atan(-1.0));
}

16
utils/bazel/llvm-project-overlay/libc/BUILD.bazel
View File

@ -1884,9 +1884,13 @@ libc_support_library(
libc_support_library(
name = "inv_trigf_utils",
srcs = ["src/math/generic/inv_trigf_utils.cpp"],
hdrs = ["src/math/generic/inv_trigf_utils.h"],
hdrs = [
"src/math/generic/inv_trigf_utils.h",
"src/math/generic/atan_utils.h",
],
deps = [
":__support_common",
":__support_fputil_double_double",
":__support_fputil_fma",
":__support_fputil_multiply_add",
":__support_fputil_polyeval",
@ -2286,6 +2290,16 @@ libc_math_function(
],
)
libc_math_function(
name = "atan",
additional_deps = [
":__support_fputil_double_double",
":__support_fputil_nearest_integer",
":__support_macros_optimization",
":inv_trigf_utils",
],
)
libc_math_function(
name = "atan2f",
additional_deps = [

2
utils/bazel/llvm-project-overlay/libc/test/src/math/BUILD.bazel
View File

@ -24,6 +24,8 @@ math_mpfr_test(name = "atan2")
math_mpfr_test(name = "atan2f")
math_mpfr_test(name = "atan")
math_mpfr_test(name = "atanf")
math_mpfr_test(name = "atanhf")

2
utils/bazel/llvm-project-overlay/libc/test/src/math/smoke/BUILD.bazel
View File

@ -26,6 +26,8 @@ math_test(name = "atan2")
math_test(name = "atan2f")
math_test(name = "atan")
math_test(name = "atanf")
math_test(name = "atanhf")