297 lines · cpp
1//===-- Double-precision tan function -------------------------------------===//2//3// Part of the LLVM Project, under the Apache License v2.0 with LLVM Exceptions.4// See https://llvm.org/LICENSE.txt for license information.5// SPDX-License-Identifier: Apache-2.0 WITH LLVM-exception6//7//===----------------------------------------------------------------------===//8 9#include "src/math/tan.h"10#include "hdr/errno_macros.h"11#include "src/__support/FPUtil/FEnvImpl.h"12#include "src/__support/FPUtil/FPBits.h"13#include "src/__support/FPUtil/PolyEval.h"14#include "src/__support/FPUtil/double_double.h"15#include "src/__support/FPUtil/dyadic_float.h"16#include "src/__support/FPUtil/except_value_utils.h"17#include "src/__support/FPUtil/multiply_add.h"18#include "src/__support/FPUtil/rounding_mode.h"19#include "src/__support/common.h"20#include "src/__support/macros/config.h"21#include "src/__support/macros/optimization.h" // LIBC_UNLIKELY22#include "src/__support/macros/properties/cpu_features.h" // LIBC_TARGET_CPU_HAS_FMA23#include "src/__support/math/range_reduction_double_common.h"24 25#ifdef LIBC_TARGET_CPU_HAS_FMA_DOUBLE26#include "src/__support/math/range_reduction_double_fma.h"27#else28#include "src/__support/math/range_reduction_double_nofma.h"29#endif // LIBC_TARGET_CPU_HAS_FMA_DOUBLE30 31namespace LIBC_NAMESPACE_DECL {32 33using DoubleDouble = fputil::DoubleDouble;34using Float128 = typename fputil::DyadicFloat<128>;35 36namespace {37 38LIBC_INLINE double tan_eval(const DoubleDouble &u, DoubleDouble &result) {39 // Evaluate tan(y) = tan(x - k * (pi/128))40 // We use the degree-9 Taylor approximation:41 // tan(y) ~ P(y) = y + y^3/3 + 2*y^5/15 + 17*y^7/315 + 62*y^9/283542 // Then the error is bounded by:43 // |tan(y) - P(y)| < 2^-6 * |y|^11 < 2^-6 * 2^-66 = 2^-72.44 // For y ~ u_hi + u_lo, fully expanding the polynomial and drop any terms45 // < ulp(u_hi^3) gives us:46 // P(y) = y + y^3/3 + 2*y^5/15 + 17*y^7/315 + 62*y^9/2835 = ...47 // ~ u_hi + u_hi^3 * (1/3 + u_hi^2 * (2/15 + u_hi^2 * (17/315 +48 // + u_hi^2 * 62/2835))) +49 // + u_lo (1 + u_hi^2 * (1 + u_hi^2 * 2/3))50 double u_hi_sq = u.hi * u.hi; // Error < ulp(u_hi^2) < 2^(-6 - 52) = 2^-58.51 // p1 ~ 17/315 + u_hi^2 62 / 2835.52 double p1 =53 fputil::multiply_add(u_hi_sq, 0x1.664f4882c10fap-6, 0x1.ba1ba1ba1ba1cp-5);54 // p2 ~ 1/3 + u_hi^2 2 / 15.55 double p2 =56 fputil::multiply_add(u_hi_sq, 0x1.1111111111111p-3, 0x1.5555555555555p-2);57 // q1 ~ 1 + u_hi^2 * 2/3.58 double q1 = fputil::multiply_add(u_hi_sq, 0x1.5555555555555p-1, 1.0);59 double u_hi_3 = u_hi_sq * u.hi;60 double u_hi_4 = u_hi_sq * u_hi_sq;61 // p3 ~ 1/3 + u_hi^2 * (2/15 + u_hi^2 * (17/315 + u_hi^2 * 62/2835))62 double p3 = fputil::multiply_add(u_hi_4, p1, p2);63 // q2 ~ 1 + u_hi^2 * (1 + u_hi^2 * 2/3)64 double q2 = fputil::multiply_add(u_hi_sq, q1, 1.0);65 double tan_lo = fputil::multiply_add(u_hi_3, p3, u.lo * q2);66 // Overall, |tan(y) - (u_hi + tan_lo)| < ulp(u_hi^3) <= 2^-71.67 // And the relative errors is:68 // |(tan(y) - (u_hi + tan_lo)) / tan(y) | <= 2*ulp(u_hi^2) < 2^-6469 result = fputil::exact_add(u.hi, tan_lo);70 return fputil::multiply_add(fputil::FPBits<double>(u_hi_3).abs().get_val(),71 0x1.0p-51, 0x1.0p-102);72}73 74#ifndef LIBC_MATH_HAS_SKIP_ACCURATE_PASS75// Accurate evaluation of tan for small u.76[[maybe_unused]] Float128 tan_eval(const Float128 &u) {77 Float128 u_sq = fputil::quick_mul(u, u);78 79 // tan(x) ~ x + x^3/3 + x^5 * 2/15 + x^7 * 17/315 + x^9 * 62/2835 +80 // + x^11 * 1382/155925 + x^13 * 21844/6081075 +81 // + x^15 * 929569/638512875 + x^17 * 6404582/1085471887582 // Relative errors < 2^-127 for |u| < pi/256.83 constexpr Float128 TAN_COEFFS[] = {84 {Sign::POS, -127, 0x80000000'00000000'00000000'00000000_u128}, // 185 {Sign::POS, -129, 0xaaaaaaaa'aaaaaaaa'aaaaaaaa'aaaaaaab_u128}, // 186 {Sign::POS, -130, 0x88888888'88888888'88888888'88888889_u128}, // 2/1587 {Sign::POS, -132, 0xdd0dd0dd'0dd0dd0d'd0dd0dd0'dd0dd0dd_u128}, // 17/31588 {Sign::POS, -133, 0xb327a441'6087cf99'6b5dd24e'ec0b327a_u128}, // 62/283589 {Sign::POS, -134,90 0x91371aaf'3611e47a'da8e1cba'7d900eca_u128}, // 1382/15592591 {Sign::POS, -136,92 0xeb69e870'abeefdaf'e606d2e4'd1e65fbc_u128}, // 21844/608107593 {Sign::POS, -137,94 0xbed1b229'5baf15b5'0ec9af45'a2619971_u128}, // 929569/63851287595 {Sign::POS, -138,96 0x9aac1240'1b3a2291'1b2ac7e3'e4627d0a_u128}, // 6404582/1085471887597 };98 99 return fputil::quick_mul(100 u, fputil::polyeval(u_sq, TAN_COEFFS[0], TAN_COEFFS[1], TAN_COEFFS[2],101 TAN_COEFFS[3], TAN_COEFFS[4], TAN_COEFFS[5],102 TAN_COEFFS[6], TAN_COEFFS[7], TAN_COEFFS[8]));103}104 105// Calculation a / b = a * (1/b) for Float128.106// Using the initial approximation of q ~ (1/b), then apply 2 Newton-Raphson107// iterations, before multiplying by a.108[[maybe_unused]] Float128 newton_raphson_div(const Float128 &a, Float128 b,109 double q) {110 Float128 q0(q);111 constexpr Float128 TWO(2.0);112 b.sign = (b.sign == Sign::POS) ? Sign::NEG : Sign::POS;113 Float128 q1 =114 fputil::quick_mul(q0, fputil::quick_add(TWO, fputil::quick_mul(b, q0)));115 Float128 q2 =116 fputil::quick_mul(q1, fputil::quick_add(TWO, fputil::quick_mul(b, q1)));117 return fputil::quick_mul(a, q2);118}119#endif // !LIBC_MATH_HAS_SKIP_ACCURATE_PASS120 121} // anonymous namespace122 123LLVM_LIBC_FUNCTION(double, tan, (double x)) {124 using namespace math::range_reduction_double_internal;125 using FPBits = typename fputil::FPBits<double>;126 FPBits xbits(x);127 128 uint16_t x_e = xbits.get_biased_exponent();129 130 DoubleDouble y;131 unsigned k;132 LargeRangeReduction range_reduction_large{};133 134 // |x| < 2^16135 if (LIBC_LIKELY(x_e < FPBits::EXP_BIAS + FAST_PASS_EXPONENT)) {136 // |x| < 2^-7137 if (LIBC_UNLIKELY(x_e < FPBits::EXP_BIAS - 7)) {138 // |x| < 2^-27, |tan(x) - x| < ulp(x)/2.139 if (LIBC_UNLIKELY(x_e < FPBits::EXP_BIAS - 27)) {140 // Signed zeros.141 if (LIBC_UNLIKELY(x == 0.0))142 return x + x; // Make sure it works with FTZ/DAZ.143 144#ifdef LIBC_TARGET_CPU_HAS_FMA_DOUBLE145 return fputil::multiply_add(x, 0x1.0p-54, x);146#else147 if (LIBC_UNLIKELY(x_e < 4)) {148 int rounding_mode = fputil::quick_get_round();149 if ((xbits.sign() == Sign::POS && rounding_mode == FE_UPWARD) ||150 (xbits.sign() == Sign::NEG && rounding_mode == FE_DOWNWARD))151 return FPBits(xbits.uintval() + 1).get_val();152 }153 return fputil::multiply_add(x, 0x1.0p-54, x);154#endif // LIBC_TARGET_CPU_HAS_FMA_DOUBLE155 }156 // No range reduction needed.157 k = 0;158 y.lo = 0.0;159 y.hi = x;160 } else {161 // Small range reduction.162 k = range_reduction_small(x, y);163 }164 } else {165 // Inf or NaN166 if (LIBC_UNLIKELY(x_e > 2 * FPBits::EXP_BIAS)) {167 if (xbits.is_signaling_nan()) {168 fputil::raise_except_if_required(FE_INVALID);169 return FPBits::quiet_nan().get_val();170 }171 // tan(+-Inf) = NaN172 if (xbits.get_mantissa() == 0) {173 fputil::set_errno_if_required(EDOM);174 fputil::raise_except_if_required(FE_INVALID);175 }176 return x + FPBits::quiet_nan().get_val();177 }178 179 // Large range reduction.180 k = range_reduction_large.fast(x, y);181 }182 183 DoubleDouble tan_y;184 [[maybe_unused]] double err = tan_eval(y, tan_y);185 186 // Look up sin(k * pi/128) and cos(k * pi/128)187#ifdef LIBC_MATH_HAS_SMALL_TABLES188 // Memory saving versions. Use 65-entry table:189 auto get_idx_dd = [](unsigned kk) -> DoubleDouble {190 unsigned idx = (kk & 64) ? 64 - (kk & 63) : (kk & 63);191 DoubleDouble ans = SIN_K_PI_OVER_128[idx];192 if (kk & 128) {193 ans.hi = -ans.hi;194 ans.lo = -ans.lo;195 }196 return ans;197 };198 DoubleDouble msin_k = get_idx_dd(k + 128);199 DoubleDouble cos_k = get_idx_dd(k + 64);200#else201 // Fast look up version, but needs 256-entry table.202 // cos(k * pi/128) = sin(k * pi/128 + pi/2) = sin((k + 64) * pi/128).203 DoubleDouble msin_k = SIN_K_PI_OVER_128[(k + 128) & 255];204 DoubleDouble cos_k = SIN_K_PI_OVER_128[(k + 64) & 255];205#endif // LIBC_MATH_HAS_SMALL_TABLES206 207 // After range reduction, k = round(x * 128 / pi) and y = x - k * (pi / 128).208 // So k is an integer and -pi / 256 <= y <= pi / 256.209 // Then tan(x) = sin(x) / cos(x)210 // = sin((k * pi/128 + y) / cos((k * pi/128 + y)211 // = (cos(y) * sin(k*pi/128) + sin(y) * cos(k*pi/128)) /212 // / (cos(y) * cos(k*pi/128) - sin(y) * sin(k*pi/128))213 // = (sin(k*pi/128) + tan(y) * cos(k*pi/128)) /214 // / (cos(k*pi/128) - tan(y) * sin(k*pi/128))215 DoubleDouble cos_k_tan_y = fputil::quick_mult(tan_y, cos_k);216 DoubleDouble msin_k_tan_y = fputil::quick_mult(tan_y, msin_k);217 218 // num_dd = sin(k*pi/128) + tan(y) * cos(k*pi/128)219 DoubleDouble num_dd = fputil::exact_add<false>(cos_k_tan_y.hi, -msin_k.hi);220 // den_dd = cos(k*pi/128) - tan(y) * sin(k*pi/128)221 DoubleDouble den_dd = fputil::exact_add<false>(msin_k_tan_y.hi, cos_k.hi);222 num_dd.lo += cos_k_tan_y.lo - msin_k.lo;223 den_dd.lo += msin_k_tan_y.lo + cos_k.lo;224 225#ifdef LIBC_MATH_HAS_SKIP_ACCURATE_PASS226 double tan_x = (num_dd.hi + num_dd.lo) / (den_dd.hi + den_dd.lo);227 return tan_x;228#else229 // Accurate test and pass for correctly rounded implementation.230 231 // Accurate double-double division232 DoubleDouble tan_x = fputil::div(num_dd, den_dd);233 234 // Simple error bound: |1 / den_dd| < 2^(1 + floor(-log2(den_dd)))).235 uint64_t den_inv = (static_cast<uint64_t>(FPBits::EXP_BIAS + 1)236 << (FPBits::FRACTION_LEN + 1)) -237 (FPBits(den_dd.hi).uintval() & FPBits::EXP_MASK);238 239 // For tan_x = (num_dd + err) / (den_dd + err), the error is bounded by:240 // | tan_x - num_dd / den_dd | <= err * ( 1 + | tan_x * den_dd | ).241 double tan_err =242 err * fputil::multiply_add(FPBits(den_inv).get_val(),243 FPBits(tan_x.hi).abs().get_val(), 1.0);244 245 double err_higher = tan_x.lo + tan_err;246 double err_lower = tan_x.lo - tan_err;247 248 double tan_upper = tan_x.hi + err_higher;249 double tan_lower = tan_x.hi + err_lower;250 251 // Ziv's rounding test.252 if (LIBC_LIKELY(tan_upper == tan_lower))253 return tan_upper;254 255 Float128 u_f128;256 if (LIBC_LIKELY(x_e < FPBits::EXP_BIAS + FAST_PASS_EXPONENT))257 u_f128 = range_reduction_small_f128(x);258 else259 u_f128 = range_reduction_large.accurate();260 261 Float128 tan_u = tan_eval(u_f128);262 263 auto get_sin_k = [](unsigned kk) -> Float128 {264 unsigned idx = (kk & 64) ? 64 - (kk & 63) : (kk & 63);265 Float128 ans = SIN_K_PI_OVER_128_F128[idx];266 if (kk & 128)267 ans.sign = Sign::NEG;268 return ans;269 };270 271 // cos(k * pi/128) = sin(k * pi/128 + pi/2) = sin((k + 64) * pi/128).272 Float128 sin_k_f128 = get_sin_k(k);273 Float128 cos_k_f128 = get_sin_k(k + 64);274 Float128 msin_k_f128 = get_sin_k(k + 128);275 276 // num_f128 = sin(k*pi/128) + tan(y) * cos(k*pi/128)277 Float128 num_f128 =278 fputil::quick_add(sin_k_f128, fputil::quick_mul(cos_k_f128, tan_u));279 // den_f128 = cos(k*pi/128) - tan(y) * sin(k*pi/128)280 Float128 den_f128 =281 fputil::quick_add(cos_k_f128, fputil::quick_mul(msin_k_f128, tan_u));282 283 // tan(x) = (sin(k*pi/128) + tan(y) * cos(k*pi/128)) /284 // / (cos(k*pi/128) - tan(y) * sin(k*pi/128))285 // TODO: The initial seed 1.0/den_dd.hi for Newton-Raphson reciprocal can be286 // reused from DoubleDouble fputil::div in the fast pass.287 Float128 result = newton_raphson_div(num_f128, den_f128, 1.0 / den_dd.hi);288 289 // TODO: Add assertion if Ziv's accuracy tests fail in debug mode.290 // https://github.com/llvm/llvm-project/issues/96452.291 return static_cast<double>(result);292 293#endif // !LIBC_MATH_HAS_SKIP_ACCURATE_PASS294}295 296} // namespace LIBC_NAMESPACE_DECL297