| //===-- 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___SUPPORT_MATH_ATAN_UTILS_H |
| #define LLVM_LIBC_SRC___SUPPORT_MATH_ATAN_UTILS_H |
| |
| #include "src/__support/FPUtil/PolyEval.h" |
| #include "src/__support/FPUtil/double_double.h" |
| #include "src/__support/FPUtil/dyadic_float.h" |
| #include "src/__support/FPUtil/multiply_add.h" |
| #include "src/__support/integer_literals.h" |
| #include "src/__support/macros/config.h" |
| |
| namespace LIBC_NAMESPACE_DECL { |
| |
| namespace atan_internal { |
| |
| using DoubleDouble = fputil::DoubleDouble; |
| using DFloat128 = fputil::DyadicFloat<128>; |
| |
| // 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, "},"); |
| // }; |
| LIBC_INLINE_VAR constexpr 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. |
| [[maybe_unused]] LIBC_INLINE 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; |
| } |
| |
| // DFloat128 versions. |
| // atan(i/64) with i = 0..64, generated by Sollya with: |
| // > for i from 1 to 64 do { |
| // a = round(atan(i/64), 128, RN); |
| // ll = ceil(log2(a)); |
| // b = 2^ll + a; |
| // print("{Sign::POS, ", 2^(ll - 128), ",", b, "},"); |
| // }; |
| LIBC_INLINE_VAR constexpr DFloat128 ATAN_I_F128[65] = { |
| {Sign::POS, 0, 0_u128}, |
| {Sign::POS, -134, 0xfffaaadd'db94d5bb'e78c5640'15f76048_u128}, |
| {Sign::POS, -133, 0xffeaaddd'4bb12542'779d776d'da8c6214_u128}, |
| {Sign::POS, -132, 0xbfdc0c21'86d14fcf'220e10d6'1df56ec7_u128}, |
| {Sign::POS, -132, 0xffaaddb9'67ef4e36'cb2792dc'0e2e0d51_u128}, |
| {Sign::POS, -131, 0x9facf873'e2aceb58'99c50bbf'08e6cdf6_u128}, |
| {Sign::POS, -131, 0xbf70c130'17887460'93567e78'4cf83676_u128}, |
| {Sign::POS, -131, 0xdf1cf5f3'783e1bef'71e5340b'30e5d9ef_u128}, |
| {Sign::POS, -131, 0xfeadd4d5'617b6e32'c897989f'3e888ef8_u128}, |
| {Sign::POS, -130, 0x8f0fd7d8'21b93725'bd375929'83a0af9a_u128}, |
| {Sign::POS, -130, 0x9eb77746'331362c3'47619d25'0360fe85_u128}, |
| {Sign::POS, -130, 0xae4c08f1'f6134efa'b54d3fef'0c2de994_u128}, |
| {Sign::POS, -130, 0xbdcbda5e'72d81134'7b0b4f88'1c9c7488_u128}, |
| {Sign::POS, -130, 0xcd35474b'643130e7'b00f3da1'a46eeb3b_u128}, |
| {Sign::POS, -130, 0xdc86ba94'93051022'f621a5c1'cb552f03_u128}, |
| {Sign::POS, -130, 0xebbeaef9'02b9b38c'91a2a68b'2fbd78e8_u128}, |
| {Sign::POS, -130, 0xfadbafc9'6406eb15'6dc79ef5'f7a217e6_u128}, |
| {Sign::POS, -129, 0x84ee2cbe'c31b12c5'c8e72197'0cabd3a3_u128}, |
| {Sign::POS, -129, 0x8c5fad18'5f8bc130'ca4748b1'bf88298d_u128}, |
| {Sign::POS, -129, 0x93c1b902'bf7a2df1'06459240'6fe1447a_u128}, |
| {Sign::POS, -129, 0x9b13b9b8'3f5e5e69'c5abb498'd27af328_u128}, |
| {Sign::POS, -129, 0xa25521b6'15784d45'43787549'88b8d9e3_u128}, |
| {Sign::POS, -129, 0xa9856cca'8e6a4eda'99b7f77b'f7d9e8c1_u128}, |
| {Sign::POS, -129, 0xb0a42018'4e7f0cb1'b51d51dc'200a0fc3_u128}, |
| {Sign::POS, -129, 0xb7b0ca0f'26f78473'8aa32122'dcfe4483_u128}, |
| {Sign::POS, -129, 0xbeab025b'1d9fbad3'910b8564'93411026_u128}, |
| {Sign::POS, -129, 0xc59269ca'50d92b6d'a1746e91'f50a28de_u128}, |
| {Sign::POS, -129, 0xcc66aa2a'6b58c33c'd9311fa1'4ed9b7c4_u128}, |
| {Sign::POS, -129, 0xd327761e'611fe5b6'427c95e9'001e7136_u128}, |
| {Sign::POS, -129, 0xd9d488ed'32e3635c'30f6394a'0806345d_u128}, |
| {Sign::POS, -129, 0xe06da64a'764f7c67'c631ed96'798cb804_u128}, |
| {Sign::POS, -129, 0xe6f29a19'609a84ba'60b77ce1'ca6dc2c8_u128}, |
| {Sign::POS, -129, 0xed63382b'0dda7b45'6fe445ec'bc3a8d03_u128}, |
| {Sign::POS, -129, 0xf3bf5bf8'bad1a21c'a7b837e6'86adf3fa_u128}, |
| {Sign::POS, -129, 0xfa06e85a'a0a0be5c'66d23c7d'5dc8ecc2_u128}, |
| {Sign::POS, -128, 0x801ce39e'0d205c99'a6d6c6c5'4d938596_u128}, |
| {Sign::POS, -128, 0x832bf4a6'd9867e2a'4b6a09cb'61a515c1_u128}, |
| {Sign::POS, -128, 0x8630a2da'da1ed065'd3e84ed5'013ca37e_u128}, |
| {Sign::POS, -128, 0x892aecdf'de9547b5'094478fc'472b4afc_u128}, |
| {Sign::POS, -128, 0x8c1ad445'f3e09b8c'439d8018'60205921_u128}, |
| {Sign::POS, -128, 0x8f005d5e'f7f59f9b'5c835e16'65c43748_u128}, |
| {Sign::POS, -128, 0x91db8f16'64f350e2'10e4f9c1'126e0220_u128}, |
| {Sign::POS, -128, 0x94ac72c9'847186f6'18c4f393'f78a32f9_u128}, |
| {Sign::POS, -128, 0x97731420'365e538b'abd3fe19'f1aeb6b3_u128}, |
| {Sign::POS, -128, 0x9a2f80e6'71bdda20'4226f8e2'204ff3bd_u128}, |
| {Sign::POS, -128, 0x9ce1c8e6'a0b8cdb9'f799c4e8'174cf11c_u128}, |
| {Sign::POS, -128, 0x9f89fdc4'f4b7a1ec'f8b49264'4f0701e0_u128}, |
| {Sign::POS, -128, 0xa22832db'cadaae08'92fe9c08'637af0e6_u128}, |
| {Sign::POS, -128, 0xa4bc7d19'34f70924'19a87f2a'457dac9f_u128}, |
| {Sign::POS, -128, 0xa746f2dd'b7602294'67b7d66f'2d74e019_u128}, |
| {Sign::POS, -128, 0xa9c7abdc'4830f5c8'916a84b5'be7933f6_u128}, |
| {Sign::POS, -128, 0xac3ec0fb'997dd6a1'a36273a5'6afa8ef4_u128}, |
| {Sign::POS, -128, 0xaeac4c38'b4d8c080'14725e2f'3e52070a_u128}, |
| {Sign::POS, -128, 0xb110688a'ebdc6f6a'43d65788'b9f6a7b5_u128}, |
| {Sign::POS, -128, 0xb36b31c9'1f043691'59014174'4462f93a_u128}, |
| {Sign::POS, -128, 0xb5bcc490'59ecc4af'f8f3cee7'5e3907d5_u128}, |
| {Sign::POS, -128, 0xb8053e2b'c2319e73'cb2da552'10a4443d_u128}, |
| {Sign::POS, -128, 0xba44bc7d'd470782f'654c2cb1'0942e386_u128}, |
| {Sign::POS, -128, 0xbc7b5dea'e98af280'd4113006'e80fb290_u128}, |
| {Sign::POS, -128, 0xbea94144'fd049aac'1043c5e7'55282e7d_u128}, |
| {Sign::POS, -128, 0xc0ce85b8'ac526640'89dd62c4'6e92fa25_u128}, |
| {Sign::POS, -128, 0xc2eb4abb'661628b5'b373fe45'c61bb9fb_u128}, |
| {Sign::POS, -128, 0xc4ffaffa'bf8fbd54'8cb43d10'bc9e0221_u128}, |
| {Sign::POS, -128, 0xc70bd54c'e602ee13'e7d54fbd'09f2be38_u128}, |
| {Sign::POS, -128, 0xc90fdaa2'2168c234'c4c6628b'80dc1cd1_u128}, |
| }; |
| |
| // Degree-13 minimax polynomial generated by Sollya with: |
| // > P = fpminimax(atan(x), [|1, 3, 5, 7, 9, 11, 13|], [|1, 128...|], |
| // [0, 2^-7]); |
| // > dirtyinfnorm(atan(x) - P, [0, 2^-7]); |
| // 0x1.26016ad97f323875760f869684c0898d7b7bb8bep-122 |
| LIBC_INLINE_VAR constexpr DFloat128 ATAN_POLY_F128[] = { |
| {Sign::NEG, -129, 0xaaaaaaaa'aaaaaaaa'aaaaaaa6'003c5d1d_u128}, |
| {Sign::POS, -130, 0xcccccccc'cccccccc'cca00232'8776b063_u128}, |
| {Sign::NEG, -130, 0x92492492'49249201'27f5268a'cb24aec0_u128}, |
| {Sign::POS, -131, 0xe38e38e3'8dce3d96'626a1643'f8eb68f3_u128}, |
| {Sign::NEG, -131, 0xba2e8b7a'ea4ad00f'005a35c7'6ef609b1_u128}, |
| {Sign::POS, -131, 0x9d82765e'd22a7d92'ac09c405'c0a69214_u128}, |
| }; |
| |
| // Approximate atan for |x| <= 2^-7. |
| [[maybe_unused]] LIBC_INLINE constexpr DFloat128 atan_eval(const DFloat128 &x) { |
| DFloat128 x_sq = fputil::quick_mul(x, x); |
| DFloat128 x3 = fputil::quick_mul(x, x_sq); |
| DFloat128 p = fputil::polyeval(x_sq, ATAN_POLY_F128[0], ATAN_POLY_F128[1], |
| ATAN_POLY_F128[2], ATAN_POLY_F128[3], |
| ATAN_POLY_F128[4], ATAN_POLY_F128[5]); |
| return fputil::multiply_add(x3, p, x); |
| } |
| |
| // Polynomial approximation for 0 <= x <= 1: |
| // atan(x) ~ atan((i/16) + (x - (i/16)) * Q(x - i/16) |
| // = P(x - i/16) |
| // Generated by Sollya with: |
| // > for i from 1 to 16 do { |
| // mid_point = i/16; |
| // P = fpminimax(atan(mid_point + x), 8, [|D...|], [-1/32, 1/32]); |
| // print("{", coeff(P, 0), ",", coeff(P, 1), ",", coeff(P, 2), ",", |
| // coeff(P, 3), ",", coeff(P, 4), ",", coeff(P, 5), ",", coeff(P, 6), |
| // ",", coeff(P, 7), ",", coeff(P, 8), "},"); |
| // }; |
| // For i = 0, the polynomial is generated by: |
| // > P = fpminimax(atan(x)/x, 7, [|1, D...|], [0, 1/32]); |
| // > dirtyinfnorm((atan(x) - x*P)/x, [0, 1/32]); |
| // 0x1.feb2fcdba66447ccbe28a1a0f935b51678a718fb1p-59 |
| // Notice that degree-7 is good enough for atanf, but degree-8 helps reduce the |
| // error bounds for atan2f's fast pass 16 times, and it does not affect the |
| // performance of atanf much. |
| LIBC_INLINE_VAR constexpr double ATAN_COEFFS[17][9] = { |
| {0.0, 1.0, 0x1.3f8d76d26d61bp-47, -0x1.5555555574cd8p-2, |
| 0x1.0dde5d06878eap-29, 0x1.99997738acc77p-3, 0x1.2c43eac9797cap-16, |
| -0x1.25fb020007dbdp-3, 0x1.c1b6c31d7b0aep-7}, |
| {0x1.ff55bb72cfde9p-5, 0x1.fe01fe01fe007p-1, -0x1.fc05f809ed8dap-5, |
| -0x1.4d69303afe04ep-2, 0x1.f61bc3e8349cp-5, 0x1.820839278756bp-3, |
| -0x1.eda4de1c6bf3fp-5, -0x1.0514d42d64a63p-3, 0x1.db3746a442dcbp-5}, |
| {0x1.fd5ba9aac2f6ep-4, 0x1.f81f81f81f813p-1, -0x1.f05e09d0dc378p-4, |
| -0x1.368c3aa719215p-2, 0x1.d9b16b33ff9c9p-4, 0x1.40488f9c6262ap-3, |
| -0x1.ba55933e62ea5p-4, -0x1.64c6a15cd9116p-4, 0x1.9273d5939a75ap-4}, |
| {0x1.7b97b4bce5b02p-3, 0x1.ee9c7f8458e05p-1, -0x1.665c226d6961p-3, |
| -0x1.1344bb7391703p-2, 0x1.42aca8b0081b9p-3, 0x1.c32d9381d7c03p-4, |
| -0x1.13e970672e246p-3, -0x1.181ed934dd733p-5, 0x1.bad81ea190c08p-4}, |
| {0x1.f5b75f92c80ddp-3, 0x1.e1e1e1e1e1e2cp-1, -0x1.c5894d10d363dp-3, |
| -0x1.ce6de025f9f5ep-3, 0x1.78a3a07c8dd7fp-3, 0x1.dd5f5180f386ep-5, |
| -0x1.1b1f513c4536bp-3, 0x1.0df852e58c43cp-6, 0x1.722e7a7e42505p-4}, |
| {0x1.362773707ebccp-2, 0x1.d272ca3fc5b2ep-1, -0x1.0997e8aeca8fbp-2, |
| -0x1.6cf6666e5e693p-3, 0x1.8dd1e907e88adp-3, 0x1.24849ac0caa5dp-7, |
| -0x1.f496be486229dp-4, 0x1.b7d54b8e759ecp-5, 0x1.d39c0d39c3922p-5}, |
| {0x1.6f61941e4def1p-2, 0x1.c0e070381c0f2p-1, -0x1.2726dd135d9eep-2, |
| -0x1.09f37b39b70e4p-3, 0x1.85eacdaadd712p-3, -0x1.04d66340d5b9p-5, |
| -0x1.8056b15a22b98p-4, 0x1.29baf494ad3ddp-4, 0x1.52d5881322a7ap-6}, |
| {0x1.a64eec3cc23fdp-2, 0x1.adbe87f94906ap-1, -0x1.3b9d8eab55addp-2, |
| -0x1.57c09646eb7p-4, 0x1.6795319e3b8dfp-3, -0x1.f2d89b5ef31bep-5, |
| -0x1.f38aac26203cap-5, 0x1.3262802235e3fp-4, -0x1.2afd6b9a57d66p-7}, |
| {0x1.dac670561bb4fp-2, 0x1.99999999999ap-1, -0x1.47ae147adff11p-2, |
| -0x1.5d867c40188b7p-5, 0x1.3a92a2df85e7ap-3, -0x1.3ec457c46e851p-4, |
| -0x1.ec1b9777e2e5bp-6, 0x1.0a542992a821ep-4, -0x1.ccffbe2f0d945p-6}, |
| {0x1.0657e94db30dp-1, 0x1.84f00c2780615p-1, -0x1.4c62cb562defap-2, |
| -0x1.e6495b3c14e03p-8, 0x1.063c2fa617bfcp-3, -0x1.58b782d9907aap-4, |
| -0x1.41e6ff524b7fp-8, 0x1.937dfff3205a7p-5, -0x1.0fb1fd1c729dp-5}, |
| {0x1.1e00babdefeb4p-1, 0x1.702e05c0b816ep-1, -0x1.4af2b78215fbep-2, |
| 0x1.5d0b7e9f36997p-6, 0x1.a1247cb978debp-4, -0x1.519e1457734cap-4, |
| 0x1.a755cf86b5bfbp-7, 0x1.096d174284564p-5, -0x1.081adf539ad58p-5}, |
| {0x1.345f01cce37bbp-1, 0x1.5babcc647fa8ep-1, -0x1.449db09426a6dp-2, |
| 0x1.655caac5896dap-5, 0x1.3bbbd22d05a61p-4, -0x1.34a2febee042fp-4, |
| 0x1.84df9c8269e34p-6, 0x1.200e8176c899ap-6, -0x1.c00b23c3ce222p-6}, |
| {0x1.4978fa3269ee1p-1, 0x1.47ae147ae1477p-1, -0x1.3a92a3055231ap-2, |
| 0x1.ec21b515a4a2p-5, 0x1.c2f8b81f9a0d2p-5, -0x1.0ba9964125453p-4, |
| 0x1.d7b5614777a05p-6, 0x1.971e91ed73595p-8, -0x1.3fc375a78dc74p-6}, |
| {0x1.5d58987169b18p-1, 0x1.34679ace01343p-1, -0x1.2ddfb039136e5p-2, |
| 0x1.2491307b9fb73p-4, 0x1.29c7e4886dc22p-5, -0x1.bca78bcca83ap-5, |
| 0x1.e63efd7cbe1ddp-6, -0x1.8ea6c4f03b42dp-10, -0x1.9385b5c3a6997p-7}, |
| {0x1.700a7c5784634p-1, 0x1.21fb78121fb76p-1, -0x1.1f6a8499e5d1ap-2, |
| 0x1.41b15e5e29423p-4, 0x1.59bc953163345p-6, -0x1.63b54b13184ddp-5, |
| 0x1.c9086666d213p-6, -0x1.90c3b4ad8d4bcp-8, -0x1.80f08ed9f6f57p-8}, |
| {0x1.819d0b7158a4dp-1, 0x1.107fbbe01107ep-1, -0x1.0feeb4089670ep-2, |
| 0x1.50e5afb93f5cbp-4, 0x1.2a7c2adffeffbp-7, -0x1.12bd29b4f1b43p-5, |
| 0x1.93f71f0eb00eap-6, -0x1.10ece5ad30e28p-7, -0x1.db1a76bcd2b9cp-10}, |
| {0x1.921fb54442d18p-1, 0x1.ffffffffffffep-2, -0x1.fffffffffc51cp-3, |
| 0x1.555555557002ep-4, -0x1.a88260c338e75p-30, -0x1.99999f9a7614fp-6, |
| 0x1.555e31a1e15e9p-6, -0x1.245240d65e629p-7, -0x1.fa9ba66478903p-11}, |
| }; |
| |
| // Look-up table for atan(k/16) with k = 0..16. |
| LIBC_INLINE_VAR constexpr double ATAN_K_OVER_16[17] = { |
| 0x0.000p+0, 0x1.ff55bb72cfdeap-5, 0x1.fd5ba9aac2f6ep-4, |
| 0x1.7b97b4bce5b02p-3, 0x1.f5b75f92c80ddp-3, 0x1.362773707ebccp-2, |
| 0x1.6f61941e4def1p-2, 0x1.a64eec3cc23fdp-2, 0x1.dac670561bb4fp-2, |
| 0x1.0657e94db30dp-1, 0x1.1e00babdefeb4p-1, 0x1.345f01cce37bbp-1, |
| 0x1.4978fa3269ee1p-1, 0x1.5d58987169b18p-1, 0x1.700a7c5784634p-1, |
| 0x1.819d0b7158a4dp-1, 0x1.921fb54442d18p-1}; |
| |
| // For |x| <= 1/32 and 0 <= i <= 16, return Q(x) such that: |
| // Q(x) ~ (atan(x + i/16) - atan(i/16)) / x. |
| LIBC_INLINE double atan_eval(double x, unsigned i) { |
| double x2 = x * x; |
| |
| double c0 = fputil::multiply_add(x, ATAN_COEFFS[i][2], ATAN_COEFFS[i][1]); |
| double c1 = fputil::multiply_add(x, ATAN_COEFFS[i][4], ATAN_COEFFS[i][3]); |
| double c2 = fputil::multiply_add(x, ATAN_COEFFS[i][6], ATAN_COEFFS[i][5]); |
| double c3 = fputil::multiply_add(x, ATAN_COEFFS[i][8], ATAN_COEFFS[i][7]); |
| |
| double x4 = x2 * x2; |
| double d1 = fputil::multiply_add(x2, c1, c0); |
| double d2 = fputil::multiply_add(x2, c3, c2); |
| double p = fputil::multiply_add(x4, d2, d1); |
| return p; |
| } |
| |
| // Evaluate atan without big lookup table. |
| // atan(n/d) - atan(k/16) = atan((n/d - k/16) / (1 + (n/d) * (k/16))) |
| // = atan((n - d * k/16)) / (d + n * k/16)) |
| // So we let q = (n - d * k/16) / (d + n * k/16), |
| // and approximate with Taylor polynomial: |
| // atan(q) ~ q - q^3/3 + q^5/5 - q^7/7 + q^9/9 |
| LIBC_INLINE double atan_eval_no_table(double num, double den, |
| double k_over_16) { |
| double num_r = fputil::multiply_add(den, -k_over_16, num); |
| double den_r = fputil::multiply_add(num, k_over_16, den); |
| double q = num_r / den_r; |
| |
| constexpr double ATAN_TAYLOR[] = { |
| -0x1.5555555555555p-2, |
| 0x1.999999999999ap-3, |
| -0x1.2492492492492p-3, |
| 0x1.c71c71c71c71cp-4, |
| }; |
| double q2 = q * q; |
| double q3 = q2 * q; |
| double q4 = q2 * q2; |
| double c0 = fputil::multiply_add(q2, ATAN_TAYLOR[1], ATAN_TAYLOR[0]); |
| double c1 = fputil::multiply_add(q2, ATAN_TAYLOR[3], ATAN_TAYLOR[2]); |
| double d = fputil::multiply_add(q4, c1, c0); |
| return fputil::multiply_add(q3, d, q); |
| } |
| |
| } // namespace atan_internal |
| |
| } // namespace LIBC_NAMESPACE_DECL |
| |
| #endif // LLVM_LIBC_SRC___SUPPORT_MATH_ATAN_UTILS_H |
