[libc][math] Implement fast pass for double precision atan function. #132333
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@llvm/pr-subscribers-libc Author: None (lntue) ChangesImplement fast pass for double precision Patch is 30.10 KiB, truncated to 20.00 KiB below, full version: https://github.com/llvm/llvm-project/pull/132333.diff 16 Files Affected:
diff --git a/libc/config/darwin/arm/entrypoints.txt b/libc/config/darwin/arm/entrypoints.txt
index 7972d285f963a..23c3ce82199fd 100644
--- a/libc/config/darwin/arm/entrypoints.txt
+++ b/libc/config/darwin/arm/entrypoints.txt
@@ -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
diff --git a/libc/config/linux/aarch64/entrypoints.txt b/libc/config/linux/aarch64/entrypoints.txt
index 431975463fbad..5c4913a658c2f 100644
--- a/libc/config/linux/aarch64/entrypoints.txt
+++ b/libc/config/linux/aarch64/entrypoints.txt
@@ -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
diff --git a/libc/config/linux/arm/entrypoints.txt b/libc/config/linux/arm/entrypoints.txt
index 170525c5529a0..9b09cc583904c 100644
--- a/libc/config/linux/arm/entrypoints.txt
+++ b/libc/config/linux/arm/entrypoints.txt
@@ -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
diff --git a/libc/config/linux/riscv/entrypoints.txt b/libc/config/linux/riscv/entrypoints.txt
index c234ea2560043..9cf05ef6d5a61 100644
--- a/libc/config/linux/riscv/entrypoints.txt
+++ b/libc/config/linux/riscv/entrypoints.txt
@@ -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
diff --git a/libc/config/linux/x86_64/entrypoints.txt b/libc/config/linux/x86_64/entrypoints.txt
index 5abe6957d2e3c..124b80d03d846 100644
--- a/libc/config/linux/x86_64/entrypoints.txt
+++ b/libc/config/linux/x86_64/entrypoints.txt
@@ -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
diff --git a/libc/config/windows/entrypoints.txt b/libc/config/windows/entrypoints.txt
index aad320995d339..319815e590425 100644
--- a/libc/config/windows/entrypoints.txt
+++ b/libc/config/windows/entrypoints.txt
@@ -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
diff --git a/libc/docs/headers/math/index.rst b/libc/docs/headers/math/index.rst
index 5b855ce4881c3..85585f4fc5f3d 100644
--- a/libc/docs/headers/math/index.rst
+++ b/libc/docs/headers/math/index.rst
@@ -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 |
+-----------+------------------+-----------------+------------------------+----------------------+------------------------+------------------------+----------------------------+
diff --git a/libc/include/math.yaml b/libc/include/math.yaml
index a66f981030864..fbfc51348411a 100644
--- a/libc/include/math.yaml
+++ b/libc/include/math.yaml
@@ -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
diff --git a/libc/src/math/generic/CMakeLists.txt b/libc/src/math/generic/CMakeLists.txt
index 3114289bad486..1a61dca07add1 100644
--- a/libc/src/math/generic/CMakeLists.txt
+++ b/libc/src/math/generic/CMakeLists.txt
@@ -4054,6 +4054,16 @@ add_entrypoint_object(
libc.src.__support.macros.properties.types
)
+add_header_library(
+ atan_util
+ 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_util
+ 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
@@ -4095,15 +4123,13 @@ add_entrypoint_object(
atan2.cpp
HDRS
../atan2.h
+ atan_utils.h
DEPENDS
- .inv_trigf_utils
+ .atan_util
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
)
diff --git a/libc/src/math/generic/atan.cpp b/libc/src/math/generic/atan.cpp
new file mode 100644
index 0000000000000..cbca60536a3fd
--- /dev/null
+++ b/libc/src/math/generic/atan.cpp
@@ -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
diff --git a/libc/src/math/generic/atan2.cpp b/libc/src/math/generic/atan2.cpp
index 1b16e15d29d0b..8adfe3321a9ee 100644
--- a/libc/src/math/generic/atan2.cpp
+++ b/libc/src/math/generic/atan2.cpp
@@ -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:
diff --git a/libc/src/math/generic/atan_utils.h b/libc/src/math/generic/atan_utils.h
new file mode 100644
index 0000000000000..3331843900d4b
--- /dev/null
+++ b/libc/src/math/generic/atan_utils.h
@@ -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.c934d...
[truncated]
|
petrhosek
reviewed
Mar 21, 2025
petrhosek
approved these changes
Mar 21, 2025
jhuber6
approved these changes
Mar 21, 2025
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Implement fast pass for double precision
atanusing range reduction modulo 1/64 and degree-9 Taylor polynomial.Relative error is bounded by 2^-66.