343 lines
14 KiB
C++
343 lines
14 KiB
C++
//===-- Single-precision atan2f function ----------------------------------===//
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//
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// Part of the LLVM Project, under the Apache License v2.0 with LLVM Exceptions.
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// See https://llvm.org/LICENSE.txt for license information.
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// SPDX-License-Identifier: Apache-2.0 WITH LLVM-exception
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//
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//===----------------------------------------------------------------------===//
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#include "src/math/atan2f.h"
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#include "hdr/fenv_macros.h"
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#include "src/__support/FPUtil/FEnvImpl.h"
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#include "src/__support/FPUtil/FPBits.h"
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#include "src/__support/FPUtil/PolyEval.h"
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#include "src/__support/FPUtil/double_double.h"
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#include "src/__support/FPUtil/multiply_add.h"
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#include "src/__support/FPUtil/nearest_integer.h"
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#include "src/__support/FPUtil/rounding_mode.h"
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#include "src/__support/macros/config.h"
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#include "src/__support/macros/optimization.h" // LIBC_UNLIKELY
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#include "src/__support/math/inv_trigf_utils.h"
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#if defined(LIBC_MATH_HAS_SKIP_ACCURATE_PASS) && \
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defined(LIBC_MATH_HAS_INTERMEDIATE_COMP_IN_FLOAT)
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// We use float-float implementation to reduce size.
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#include "src/math/generic/atan2f_float.h"
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#else
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namespace LIBC_NAMESPACE_DECL {
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namespace {
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#ifndef LIBC_MATH_HAS_SKIP_ACCURATE_PASS
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// Look up tables for accurate pass:
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// atan(i/16) with i = 0..16, generated by Sollya with:
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// > for i from 0 to 16 do {
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// a = round(atan(i/16), D, RN);
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// b = round(atan(i/16) - a, D, RN);
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// print("{", b, ",", a, "},");
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// };
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constexpr fputil::DoubleDouble ATAN_I[17] = {
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{0.0, 0.0},
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{-0x1.c934d86d23f1dp-60, 0x1.ff55bb72cfdeap-5},
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{-0x1.cd37686760c17p-59, 0x1.fd5ba9aac2f6ep-4},
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{0x1.347b0b4f881cap-58, 0x1.7b97b4bce5b02p-3},
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{0x1.8ab6e3cf7afbdp-57, 0x1.f5b75f92c80ddp-3},
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{-0x1.963a544b672d8p-57, 0x1.362773707ebccp-2},
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{-0x1.c63aae6f6e918p-56, 0x1.6f61941e4def1p-2},
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{-0x1.24dec1b50b7ffp-56, 0x1.a64eec3cc23fdp-2},
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{0x1.a2b7f222f65e2p-56, 0x1.dac670561bb4fp-2},
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{-0x1.d5b495f6349e6p-56, 0x1.0657e94db30dp-1},
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{-0x1.928df287a668fp-58, 0x1.1e00babdefeb4p-1},
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{0x1.1021137c71102p-55, 0x1.345f01cce37bbp-1},
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{0x1.2419a87f2a458p-56, 0x1.4978fa3269ee1p-1},
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{0x1.0028e4bc5e7cap-57, 0x1.5d58987169b18p-1},
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{-0x1.8c34d25aadef6p-56, 0x1.700a7c5784634p-1},
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{-0x1.bf76229d3b917p-56, 0x1.819d0b7158a4dp-1},
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{0x1.1a62633145c07p-55, 0x1.921fb54442d18p-1},
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};
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// Taylor polynomial, generated by Sollya with:
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// > for i from 0 to 8 do {
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// j = (-1)^(i + 1)/(2*i + 1);
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// a = round(j, D, RN);
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// b = round(j - a, D, RN);
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// print("{", b, ",", a, "},");
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// };
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constexpr fputil::DoubleDouble COEFFS[9] = {
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{0.0, 1.0}, // 1
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{-0x1.5555555555555p-56, -0x1.5555555555555p-2}, // -1/3
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{-0x1.999999999999ap-57, 0x1.999999999999ap-3}, // 1/5
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{-0x1.2492492492492p-57, -0x1.2492492492492p-3}, // -1/7
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{0x1.c71c71c71c71cp-58, 0x1.c71c71c71c71cp-4}, // 1/9
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{0x1.745d1745d1746p-59, -0x1.745d1745d1746p-4}, // -1/11
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{-0x1.3b13b13b13b14p-58, 0x1.3b13b13b13b14p-4}, // 1/13
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{-0x1.1111111111111p-60, -0x1.1111111111111p-4}, // -1/15
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{0x1.e1e1e1e1e1e1ep-61, 0x1.e1e1e1e1e1e1ep-5}, // 1/17
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};
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// Veltkamp's splitting of a double precision into hi + lo, where the hi part is
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// slightly smaller than an even split, so that the product of
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// hi * (s1 * k + s2) is exact,
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// where:
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// s1, s2 are single precsion,
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// 1/16 <= s1/s2 <= 1
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// 1/16 <= k <= 1 is an integer.
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// So the maximal precision of (s1 * k + s2) is:
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// prec(s1 * k + s2) = 2 + log2(msb(s2)) - log2(lsb(k_d * s1))
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// = 2 + log2(msb(s1)) + 4 - log2(lsb(k_d)) - log2(lsb(s1))
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// = 2 + log2(lsb(s1)) + 23 + 4 - (-4) - log2(lsb(s1))
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// = 33.
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// Thus, the Veltkamp splitting constant is C = 2^33 + 1.
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// This is used when FMA instruction is not available.
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[[maybe_unused]] constexpr fputil::DoubleDouble split_d(double a) {
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fputil::DoubleDouble r{0.0, 0.0};
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constexpr double C = 0x1.0p33 + 1.0;
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double t1 = C * a;
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double t2 = a - t1;
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r.hi = t1 + t2;
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r.lo = a - r.hi;
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return r;
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}
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// Compute atan( num_d / den_d ) in double-double precision.
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// num_d = min(|x|, |y|)
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// den_d = max(|x|, |y|)
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// q_d = num_d / den_d
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// idx, k_d = round( 2^4 * num_d / den_d )
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// final_sign = sign of the final result
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// const_term = the constant term in the final expression.
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float atan2f_double_double(double num_d, double den_d, double q_d, int idx,
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double k_d, double final_sign,
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const fputil::DoubleDouble &const_term) {
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fputil::DoubleDouble q;
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double num_r, den_r;
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if (idx != 0) {
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// The following range reduction is accurate even without fma for
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// 1/16 <= n/d <= 1.
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// atan(n/d) - atan(idx/16) = atan((n/d - idx/16) / (1 + (n/d) * (idx/16)))
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// = atan((n - d*(idx/16)) / (d + n*idx/16))
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k_d *= 0x1.0p-4;
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num_r = fputil::multiply_add(k_d, -den_d, num_d); // Exact
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den_r = fputil::multiply_add(k_d, num_d, den_d); // Exact
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q.hi = num_r / den_r;
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} else {
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// For 0 < n/d < 1/16, we just need to calculate the lower part of their
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// quotient.
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q.hi = q_d;
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num_r = num_d;
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den_r = den_d;
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}
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#ifdef LIBC_TARGET_CPU_HAS_FMA_DOUBLE
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q.lo = fputil::multiply_add(q.hi, -den_r, num_r) / den_r;
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#else
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// Compute `(num_r - q.hi * den_r) / den_r` accurately without FMA
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// instructions.
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fputil::DoubleDouble q_hi_dd = split_d(q.hi);
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double t1 = fputil::multiply_add(q_hi_dd.hi, -den_r, num_r); // Exact
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double t2 = fputil::multiply_add(q_hi_dd.lo, -den_r, t1);
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q.lo = t2 / den_r;
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#endif // LIBC_TARGET_CPU_HAS_FMA_DOUBLE
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// Taylor polynomial, evaluating using Horner's scheme:
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// P = x - x^3/3 + x^5/5 -x^7/7 + x^9/9 - x^11/11 + x^13/13 - x^15/15
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// + x^17/17
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// = x*(1 + x^2*(-1/3 + x^2*(1/5 + x^2*(-1/7 + x^2*(1/9 + x^2*
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// *(-1/11 + x^2*(1/13 + x^2*(-1/15 + x^2 * 1/17))))))))
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fputil::DoubleDouble q2 = fputil::quick_mult(q, q);
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fputil::DoubleDouble p_dd =
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fputil::polyeval(q2, COEFFS[0], COEFFS[1], COEFFS[2], COEFFS[3],
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COEFFS[4], COEFFS[5], COEFFS[6], COEFFS[7], COEFFS[8]);
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fputil::DoubleDouble r_dd =
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fputil::add(const_term, fputil::multiply_add(q, p_dd, ATAN_I[idx]));
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r_dd.hi *= final_sign;
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r_dd.lo *= final_sign;
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// Make sure the sum is normalized:
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fputil::DoubleDouble rr = fputil::exact_add(r_dd.hi, r_dd.lo);
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// Round to odd.
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uint64_t rr_bits = cpp::bit_cast<uint64_t>(rr.hi);
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if (LIBC_UNLIKELY(((rr_bits & 0xfff'ffff) == 0) && (rr.lo != 0.0))) {
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Sign hi_sign = fputil::FPBits<double>(rr.hi).sign();
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Sign lo_sign = fputil::FPBits<double>(rr.lo).sign();
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if (hi_sign == lo_sign) {
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++rr_bits;
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} else if ((rr_bits & fputil::FPBits<double>::FRACTION_MASK) > 0) {
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--rr_bits;
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}
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}
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return static_cast<float>(cpp::bit_cast<double>(rr_bits));
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}
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#endif // !LIBC_MATH_HAS_SKIP_ACCURATE_PASS
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} // anonymous namespace
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// There are several range reduction steps we can take for atan2(y, x) as
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// follow:
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// * Range reduction 1: signness
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// atan2(y, x) will return a number between -PI and PI representing the angle
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// forming by the 0x axis and the vector (x, y) on the 0xy-plane.
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// In particular, we have that:
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// atan2(y, x) = atan( y/x ) if x >= 0 and y >= 0 (I-quadrant)
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// = pi + atan( y/x ) if x < 0 and y >= 0 (II-quadrant)
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// = -pi + atan( y/x ) if x < 0 and y < 0 (III-quadrant)
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// = atan( y/x ) if x >= 0 and y < 0 (IV-quadrant)
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// Since atan function is odd, we can use the formula:
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// atan(-u) = -atan(u)
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// to adjust the above conditions a bit further:
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// atan2(y, x) = atan( |y|/|x| ) if x >= 0 and y >= 0 (I-quadrant)
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// = pi - atan( |y|/|x| ) if x < 0 and y >= 0 (II-quadrant)
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// = -pi + atan( |y|/|x| ) if x < 0 and y < 0 (III-quadrant)
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// = -atan( |y|/|x| ) if x >= 0 and y < 0 (IV-quadrant)
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// Which can be simplified to:
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// atan2(y, x) = sign(y) * atan( |y|/|x| ) if x >= 0
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// = sign(y) * (pi - atan( |y|/|x| )) if x < 0
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// * Range reduction 2: reciprocal
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// Now that the argument inside atan is positive, we can use the formula:
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// atan(1/x) = pi/2 - atan(x)
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// to make the argument inside atan <= 1 as follow:
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// atan2(y, x) = sign(y) * atan( |y|/|x|) if 0 <= |y| <= x
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// = sign(y) * (pi/2 - atan( |x|/|y| ) if 0 <= x < |y|
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// = sign(y) * (pi - atan( |y|/|x| )) if 0 <= |y| <= -x
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// = sign(y) * (pi/2 + atan( |x|/|y| )) if 0 <= -x < |y|
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// * Range reduction 3: look up table.
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// After the previous two range reduction steps, we reduce the problem to
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// compute atan(u) with 0 <= u <= 1, or to be precise:
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// atan( n / d ) where n = min(|x|, |y|) and d = max(|x|, |y|).
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// An accurate polynomial approximation for the whole [0, 1] input range will
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// require a very large degree. To make it more efficient, we reduce the input
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// range further by finding an integer idx such that:
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// | n/d - idx/16 | <= 1/32.
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// In particular,
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// idx := 2^-4 * round(2^4 * n/d)
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// Then for the fast pass, we find a polynomial approximation for:
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// atan( n/d ) ~ atan( idx/16 ) + (n/d - idx/16) * Q(n/d - idx/16)
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// For the accurate pass, we use the addition formula:
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// atan( n/d ) - atan( idx/16 ) = atan( (n/d - idx/16)/(1 + (n*idx)/(16*d)) )
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// = atan( (n - d * idx/16)/(d + n * idx/16) )
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// And finally we use Taylor polynomial to compute the RHS in the accurate pass:
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// atan(u) ~ P(u) = u - u^3/3 + u^5/5 - u^7/7 + u^9/9 - u^11/11 + u^13/13 -
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// - u^15/15 + u^17/17
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// It's error in double-double precision is estimated in Sollya to be:
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// > P = x - x^3/3 + x^5/5 -x^7/7 + x^9/9 - x^11/11 + x^13/13 - x^15/15
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// + x^17/17;
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// > dirtyinfnorm(atan(x) - P, [-2^-5, 2^-5]);
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// 0x1.aec6f...p-100
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// which is about rounding errors of double-double (2^-104).
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LLVM_LIBC_FUNCTION(float, atan2f, (float y, float x)) {
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using namespace inv_trigf_utils_internal;
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using FPBits = typename fputil::FPBits<float>;
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constexpr double IS_NEG[2] = {1.0, -1.0};
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constexpr double PI = 0x1.921fb54442d18p1;
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constexpr double PI_LO = 0x1.1a62633145c07p-53;
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constexpr double PI_OVER_4 = 0x1.921fb54442d18p-1;
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constexpr double PI_OVER_2 = 0x1.921fb54442d18p0;
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constexpr double THREE_PI_OVER_4 = 0x1.2d97c7f3321d2p+1;
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// Adjustment for constant term:
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// CONST_ADJ[x_sign][y_sign][recip]
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constexpr fputil::DoubleDouble CONST_ADJ[2][2][2] = {
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{{{0.0, 0.0}, {-PI_LO / 2, -PI_OVER_2}},
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{{-0.0, -0.0}, {-PI_LO / 2, -PI_OVER_2}}},
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{{{-PI_LO, -PI}, {PI_LO / 2, PI_OVER_2}},
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{{-PI_LO, -PI}, {PI_LO / 2, PI_OVER_2}}}};
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FPBits x_bits(x), y_bits(y);
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bool x_sign = x_bits.sign().is_neg();
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bool y_sign = y_bits.sign().is_neg();
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x_bits.set_sign(Sign::POS);
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y_bits.set_sign(Sign::POS);
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uint32_t x_abs = x_bits.uintval();
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uint32_t y_abs = y_bits.uintval();
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uint32_t max_abs = x_abs > y_abs ? x_abs : y_abs;
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uint32_t min_abs = x_abs <= y_abs ? x_abs : y_abs;
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float num_f = FPBits(min_abs).get_val();
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float den_f = FPBits(max_abs).get_val();
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double num_d = static_cast<double>(num_f);
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double den_d = static_cast<double>(den_f);
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if (LIBC_UNLIKELY(max_abs >= 0x7f80'0000U || num_d == 0.0)) {
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if (x_bits.is_nan() || y_bits.is_nan()) {
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if (x_bits.is_signaling_nan() || y_bits.is_signaling_nan())
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fputil::raise_except_if_required(FE_INVALID);
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return FPBits::quiet_nan().get_val();
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}
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double x_d = static_cast<double>(x);
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double y_d = static_cast<double>(y);
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size_t x_except = (x_d == 0.0) ? 0 : (x_abs == 0x7f80'0000 ? 2 : 1);
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size_t y_except = (y_d == 0.0) ? 0 : (y_abs == 0x7f80'0000 ? 2 : 1);
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// Exceptional cases:
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// EXCEPT[y_except][x_except][x_is_neg]
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// with x_except & y_except:
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// 0: zero
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// 1: finite, non-zero
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// 2: infinity
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constexpr double EXCEPTS[3][3][2] = {
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{{0.0, PI}, {0.0, PI}, {0.0, PI}},
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{{PI_OVER_2, PI_OVER_2}, {0.0, 0.0}, {0.0, PI}},
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{{PI_OVER_2, PI_OVER_2},
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{PI_OVER_2, PI_OVER_2},
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{PI_OVER_4, THREE_PI_OVER_4}},
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};
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double r = IS_NEG[y_sign] * EXCEPTS[y_except][x_except][x_sign];
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return static_cast<float>(r);
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}
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bool recip = x_abs < y_abs;
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double final_sign = IS_NEG[(x_sign != y_sign) != recip];
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fputil::DoubleDouble const_term = CONST_ADJ[x_sign][y_sign][recip];
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double q_d = num_d / den_d;
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double k_d = fputil::nearest_integer(q_d * 0x1.0p4);
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int idx = static_cast<int>(k_d);
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double r;
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#ifdef LIBC_MATH_HAS_SMALL_TABLES
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double p = atan_eval_no_table(num_d, den_d, k_d * 0x1.0p-4);
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r = final_sign * (p + (const_term.hi + ATAN_K_OVER_16[idx]));
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#else
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q_d = fputil::multiply_add(k_d, -0x1.0p-4, q_d);
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double p = atan_eval(q_d, idx);
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r = final_sign *
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fputil::multiply_add(q_d, p, const_term.hi + ATAN_COEFFS[idx][0]);
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#endif // LIBC_MATH_HAS_SMALL_TABLES
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#ifdef LIBC_MATH_HAS_SKIP_ACCURATE_PASS
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return static_cast<float>(r);
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#else
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constexpr uint32_t LOWER_ERR = 4;
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// Mask sticky bits in double precision before rounding to single precision.
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constexpr uint32_t MASK =
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mask_trailing_ones<uint32_t, fputil::FPBits<double>::SIG_LEN -
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FPBits::SIG_LEN - 1>();
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constexpr uint32_t UPPER_ERR = MASK - LOWER_ERR;
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uint32_t r_bits = static_cast<uint32_t>(cpp::bit_cast<uint64_t>(r)) & MASK;
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// Ziv's rounding test.
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if (LIBC_LIKELY(r_bits > LOWER_ERR && r_bits < UPPER_ERR))
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return static_cast<float>(r);
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return atan2f_double_double(num_d, den_d, q_d, idx, k_d, final_sign,
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const_term);
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#endif // LIBC_MATH_HAS_SKIP_ACCURATE_PASS
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}
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} // namespace LIBC_NAMESPACE_DECL
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#endif
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