[libc-commits] [libc] [libc][math] Implement atan2f correctly rounded to all rounding modes. (PR #86716)
Nick Desaulniers via libc-commits
libc-commits at lists.llvm.org
Tue Mar 26 13:08:23 PDT 2024
================
@@ -0,0 +1,297 @@
+//===-- Single-precision atan2f 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/atan2f.h"
+#include "inv_trigf_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/except_value_utils.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/optimization.h" // LIBC_UNLIKELY
+
+namespace LIBC_NAMESPACE {
+
+namespace {
+
+// Look up tables for accurate pass:
+
+// atan(i/16) with i = 0..16, generated by Sollya with:
+// > for i from 0 to 16 do {
+// a = round(atan(i/16), D, RN);
+// b = round(atan(i/16) - a, D, RN);
+// print("{", b, ",", a, "},");
+// };
+constexpr fputil::DoubleDouble ATAN_I[17] = {
+ {0.0, 0.0},
+ {-0x1.c934d86d23f1dp-60, 0x1.ff55bb72cfdeap-5},
+ {-0x1.cd37686760c17p-59, 0x1.fd5ba9aac2f6ep-4},
+ {0x1.347b0b4f881cap-58, 0x1.7b97b4bce5b02p-3},
+ {0x1.8ab6e3cf7afbdp-57, 0x1.f5b75f92c80ddp-3},
+ {-0x1.963a544b672d8p-57, 0x1.362773707ebccp-2},
+ {-0x1.c63aae6f6e918p-56, 0x1.6f61941e4def1p-2},
+ {-0x1.24dec1b50b7ffp-56, 0x1.a64eec3cc23fdp-2},
+ {0x1.a2b7f222f65e2p-56, 0x1.dac670561bb4fp-2},
+ {-0x1.d5b495f6349e6p-56, 0x1.0657e94db30dp-1},
+ {-0x1.928df287a668fp-58, 0x1.1e00babdefeb4p-1},
+ {0x1.1021137c71102p-55, 0x1.345f01cce37bbp-1},
+ {0x1.2419a87f2a458p-56, 0x1.4978fa3269ee1p-1},
+ {0x1.0028e4bc5e7cap-57, 0x1.5d58987169b18p-1},
+ {-0x1.8c34d25aadef6p-56, 0x1.700a7c5784634p-1},
+ {-0x1.bf76229d3b917p-56, 0x1.819d0b7158a4dp-1},
+ {0x1.1a62633145c07p-55, 0x1.921fb54442d18p-1},
+};
+
+// Taylor polynomial, generated by Sollya with:
+// > for i from 0 to 8 do {
+// j = (-1)^(i + 1)/(2*i + 1);
+// a = round(j, D, RN);
+// b = round(j - a, D, RN);
+// print("{", b, ",", a, "},");
+// };
+constexpr fputil::DoubleDouble COEFFS[9] = {
+ {0.0, 1.0}, // 1
+ {-0x1.5555555555555p-56, -0x1.5555555555555p-2}, // -1/3
+ {-0x1.999999999999ap-57, 0x1.999999999999ap-3}, // 1/5
+ {-0x1.2492492492492p-57, -0x1.2492492492492p-3}, // -1/7
+ {0x1.c71c71c71c71cp-58, 0x1.c71c71c71c71cp-4}, // 1/9
+ {0x1.745d1745d1746p-59, -0x1.745d1745d1746p-4}, // -1/11
+ {-0x1.3b13b13b13b14p-58, 0x1.3b13b13b13b14p-4}, // 1/13
+ {-0x1.1111111111111p-60, -0x1.1111111111111p-4}, // -1/15
+ {0x1.e1e1e1e1e1e1ep-61, 0x1.e1e1e1e1e1e1ep-5}, // 1/17
+};
+
+// Veltkamp's splitting of a double precision into hi + lo, where the hi part is
+// slightly smaller than an even split, so that the product of
+// hi * s * k is exact,
+// where:
+// s is single precsion,
+// 0 < k < 16 is an integer.
+// This is used when FMA instruction is not available.
+[[maybe_unused]] LIBC_INLINE constexpr fputil::DoubleDouble split_d(double a) {
+ fputil::DoubleDouble r{0.0, 0.0};
+ constexpr double C = 0x1.0p33 + 1.0;
+ double t1 = C * a;
+ double t2 = a - t1;
+ r.hi = t1 + t2;
+ r.lo = a - r.hi;
+ return r;
+}
+
+} // anonymous namespace
+
+// There are several range reduction steps we can take for atan2(y, x) as
+// follow:
+
+// * Range reduction 1: signness
+// atan2(y, x) will return a number between -PI and PI representing the angle
+// forming by the 0x axis and the vector (x, y) on the 0xy-plane.
+// In particular, we have that:
+// atan2(y, x) = atan( y/x ) if x >= 0 and y >= 0 (I-quadrant)
+// = pi + atan( y/x ) if x < 0 and y >= 0 (II-quadrant)
+// = -pi + atan( y/x ) if x < 0 and y < 0 (III-quadrant)
+// = atan( y/x ) if x >= 0 and y < 0 (IV-quadrant)
+// Since atan function is odd, we can use the formula:
+// atan(-u) = -atan(u)
+// to adjust the above conditions a bit further:
+// atan2(y, x) = atan( |y|/|x| ) if x >= 0 and y >= 0 (I-quadrant)
+// = pi - atan( |y|/|x| ) if x < 0 and y >= 0 (II-quadrant)
+// = -pi + atan( |y|/|x| ) if x < 0 and y < 0 (III-quadrant)
+// = -atan( |y|/|x| ) if x >= 0 and y < 0 (IV-quadrant)
+// Which can be simplified to:
+// atan2(y, x) = sign(y) * atan( |y|/|x| ) if x >= 0
+// = sign(y) * (pi - atan( |y|/|x| )) if x < 0
+
+// * Range reduction 2: reciprocal
+// Now that the argument inside atan is positive, we can use the formula:
+// atan(1/x) = pi/2 - atan(x)
+// to make the argument inside atan <= 1 as follow:
+// atan2(y, x) = sign(y) * atan( |y|/|x|) if 0 <= |y| <= x
+// = sign(y) * (pi/2 - atan( |x|/|y| ) if 0 <= x < |y|
+// = sign(y) * (pi - atan( |y|/|x| )) if 0 <= |y| <= -x
+// = sign(y) * (pi/2 + atan( |x|/|y| )) if 0 <= -x < |y|
+
+// * Range reduction 3: look up table.
+// After the previous two range reduction steps, we reduce the problem to
+// compute atan(u) with 0 <= u <= 1, or to be precise:
+// atan( n / d ) where n = min(|x|, |y|) and d = max(|x|, |y|).
+// An accurate polynomial approximation for the whole [0, 1] input range will
+// require a very large degree. To make it more efficient, we reduce the input
+// range further by finding an integer idx such that:
+// | n/d - idx/16 | <= 1/32.
+// In particular,
+// idx := 2^-4 * round(2^4 * n/d)
+// Then for the fast pass, we find a polynomial approximation for:
+// atan( n/d ) ~ atan( idx/16 ) + (n/d - idx/16) * Q(n/d - idx/16)
+// For the accurate pass, we use the addition formula:
+// atan( n/d ) - atan( idx/16 ) = atan( (n/d - idx/16)/(1 + (n*idx)/(16*d)) )
+// = atan( (n - d * idx/16)/(d + n * idx/16) )
+// And finally we use Taylor polynomial to compute the RHS in the accurate pass:
+// atan(u) ~ P(u) = u - u^3/3 + u^5/5 - u^7/7 + u^9/9 - u^11/11 + u^13/13 -
+// - u^15/15 + u^17/17
+// It's error in double-double precision is estimated in Sollya to be:
+// > P = x - x^3/3 + x^5/5 -x^7/7 + x^9/9 - x^11/11 + x^13/13 - x^15/15
+// + x^17/17;
+// > dirtyinfnorm(atan(x) - P, [-2^-5, 2^-5]);
+// 0x1.aec6f...p-100
+// which is about rounding errors of double-double (2^-104).
+
+LLVM_LIBC_FUNCTION(float, atan2f, (float y, float x)) {
+ using FPBits = typename fputil::FPBits<float>;
+ constexpr double IS_NEG[2] = {1.0, -1.0};
+ constexpr double PI = 0x1.921fb54442d18p1;
+ constexpr double PI_LO = 0x1.1a62633145c07p-53;
+ constexpr double PI_OVER_4 = 0x1.921fb54442d18p-1;
+ constexpr double PI_OVER_2 = 0x1.921fb54442d18p0;
+ constexpr double THREE_PI_OVER_4 = 0x1.2d97c7f3321d2p+1;
+ // Adjustment for constant term:
+ // CONST_ADJ[x_sign][y_sign][recip]
+ constexpr fputil::DoubleDouble CONST_ADJ[2][2][2] = {
+ {{{0.0, 0.0}, {-PI_LO / 2, -PI_OVER_2}},
+ {{-0.0, -0.0}, {-PI_LO / 2, -PI_OVER_2}}},
+ {{{-PI_LO, -PI}, {PI_LO / 2, PI_OVER_2}},
+ {{-PI_LO, -PI}, {PI_LO / 2, PI_OVER_2}}}};
+
+ FPBits x_bits(x), y_bits(y);
+ bool x_sign = x_bits.sign().is_neg();
+ bool y_sign = y_bits.sign().is_neg();
+ x_bits.set_sign(Sign::POS);
+ y_bits.set_sign(Sign::POS);
+ uint32_t x_abs = x_bits.uintval();
+ uint32_t y_abs = y_bits.uintval();
+ uint32_t max_abs = x_abs > y_abs ? x_abs : y_abs;
+ uint32_t min_abs = x_abs <= y_abs ? x_abs : y_abs;
+ bool recip = x_abs < y_abs;
+
+ if (LIBC_UNLIKELY(max_abs >= 0x7f80'0000U || min_abs == 0U)) {
+ if (x_bits.is_nan() || y_bits.is_nan())
+ return FPBits::quiet_nan().get_val();
+ size_t x_except = x_abs == 0 ? 0 : (x_abs == 0x7f80'0000 ? 2 : 1);
+ size_t y_except = y_abs == 0 ? 0 : (y_abs == 0x7f80'0000 ? 2 : 1);
+
+ // Exceptional cases:
+ // EXCEPT[y_except][x_except][x_is_neg]
+ // with x_except & y_except:
+ // 0: zero
+ // 1: finite, non-zero
+ // 2: infinity
+ constexpr double EXCEPTS[3][3][2] = {
+ {{0.0, PI}, {0.0, PI}, {0.0, PI}},
+ {{PI_OVER_2, PI_OVER_2}, {0.0, 0.0}, {0.0, PI}},
+ {{PI_OVER_2, PI_OVER_2},
+ {PI_OVER_2, PI_OVER_2},
+ {PI_OVER_4, THREE_PI_OVER_4}},
+ };
+
+ double r = IS_NEG[y_sign] * EXCEPTS[y_except][x_except][x_sign];
+
+ return static_cast<float>(r);
+ }
+
+ double final_sign = IS_NEG[(x_sign != y_sign) != recip];
+ fputil::DoubleDouble const_term = CONST_ADJ[x_sign][y_sign][recip];
+ double num_d = static_cast<double>(FPBits(min_abs).get_val());
+ double den_d = static_cast<double>(FPBits(max_abs).get_val());
+ double q_d = num_d / den_d;
+ int idx;
+
+ double k_d = fputil::nearest_integer(q_d * 0x1.0p4f);
+ idx = static_cast<int>(k_d);
----------------
nickdesaulniers wrote:
Combine declaration + assignment of `idx`.
https://github.com/llvm/llvm-project/pull/86716
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